The FA Cup:

Burnley 3 Middlesbrough 1, third round, February 1913. Woodrow Wilson had just been inaugurated as US president, and Richard Nixon and Danny Kay were just a few days old.

Boro were 14th in Division One, Burnley second in Division Two. Boro had reached the third round (the equivalent of today's fifth) by beating Southern League sides Millwall and QPR. But they had just lost three games in a row, letting in 13 goals. Burnley had won 14 of the their last 16 games, including a club record 10 wins in a row, and had only just been knocked off the top of Division Two. 
Special trains brought thousands of Boro fans via a torturous route along now long-vanished lines, making it the biggest crowd of the season at Turf Moor, 27,824.

Boro took the lead after 20 minutes with a goal by left-winger Edmund Eyre, but Burnley equalised 15 minutes later through Teddy Hodgson.    
The second half was described as  "heated and frantic," but Burnley went in front just before the hour through Bert Freeman, George Elliott's rival for the England centre-forward spot, and a man who'd score 36 times that season. Elliott had won his first England cap just a week earlier, which may have been why 10 minutes from time fired-up Freeman scored the decisive third on the break and Boro were out.

The Aftermath: Burnley went on to win at Blackburn in an even more heated game, and reached to the semi-finals where they went out to Sunderland after a replay. But they stayed second in the table and were promoted after 12 years in Division Two.
Boro responded by winning 3-2 at Manchester United, but only won one more game all season, and finished 16th.   A year later they would finish third, their highest-ever position.

Middlesbrough 1 Burnley 1, sixth round, March 1947.

Lord Mountbatten had just been appointed Viceroy of India, and Harry Redknapp was a few hours old.  Sadly his parents didn't have a car so they couldn't roll down the windows for pictures and a quick gurgle as they left the hospital.

Burnley were again second in Division Two, but this time Boro were second in Division One.
Burnley had lost only one of their last 20 games (just as in 1913 it was at Nottingham Forest, something all Boro fans can relate to).     
Boro had beaten Forest 6-2 in the fifth round, with Wilf Mannion getting a hat-trick, after wins over QPR (again) and Chesterfield. In the league they had just beaten Derby 1-0 at home to go back into second place after a Typical Boro New Year in which they lost three games out of four, including two 4-2 home defeats.    

It was only the second time in nearly half a century they'd reached the quarter-finals, in 1947, and a record 53,025 filled Ayresome Park. It remains the biggest cup crowd and the second-biggest ever for a Middlesbrough home game.

Geoff Walker put Boro ahead just before half-time, and when Micky Fenton smashed in a 30-yard free-kick with only seven minutes left it looked like a first-ever semi-final. But a linesman kept waving his flag as the teams prepared to kick off, and the referee eventually disallowed the goal, as Johnny Spuhler had run forward into an offside position, even though it was way out on the far wing, where the referee decided he was somehow interfering with play.
This was the same rule, interpreted the opposite way, that caused outrage in Leeds in 1971 when West Brom broke to score the winning goal despite a player standing a yard inside their half on the other side of the field. He was ruled to be not interfering, and it cost Leeds the title. Burnley took the free kick, broke away and with their only attack of the half Billy Morris equalised after Boro centre-half Norman Robinson slipped on the icy surface.

Burnley 1 Middlesbrough 0 (after extra time), sixth round replay.

If Boro felt hard done by, it was nothing to what they felt after the replay three days later. Turf Moor was frozen solid, and dangerous, said many of the players, but after the groundstaff had smashed it with pickaxes and crowbars the referee insisted it went ahead. The fact that nearly 50,000 had turned up may have had something to do with it. The only goal came in the third minute of extra time when keeper Dave Cumming went to collect a low free kick (only given because Dicky Robinson had slid into Peter Kippax on the ice) and got a boot in the face from Burnley's Ray Harrison, who then sat on him and flicked the ball across goal -  with his hand. Billy Morris turned it in, players on both sides stopped and laughed, but the referee, again after consulting the linesman gave a goal.

George Hardwick called it criminal, and admitted he should have been sent off for his long, expletive-filled protests, something unheard of from Gentleman George. Wilf Mannion was bitter about it until the day he died.
Manager David Jack went even further than even Neil Warnock would dream of, saying he didn't wish the referee any harm but he hoped he'd break his leg.

It even came back to haunt Boro a quarter of a century later. The referee was Arthur Ellis, and Boro fans had to endure 13 years watching him hamming it up on TV's It's a Knockout. Fittingly, said Hardwick, Ellis only came on when someone was playing their joker. In the long run he did of course prove to be far less notorious than one of his fellow presenters, Stuart Hall.
"The Joker" was England's top ref, doing three World Cups and the first European Cup final, incredibly managing to get his brother, who wasn't even a league referee, to run the line in all his earlier ties. He was replaced for the final - by a man called Tommy Cooper. Ellis still had problems with linesmen though - he caused a near riot when he ignored an offside flag to give Barcelona a match-saving penalty at Real Madrid.     

The Aftermath: The first game was also a PR disaster, and close to a real one. No all-ticket games in those days, so 5,000 were locked out. Those who couldn't get in when the gates were locked at 2.30 tried to storm the main entrance, and the police had to park a fire engine across the Warwick Street gates, with its water cannon punting ominously at the crowd. Some had been queueing since they opened an hour earlier. The Chief Constable calmed them down and gave a running commentary via a loud hailer. His comments on the referee are unrecorded.
Angry letters to the Gazette flooded in all week, but there was also praise for a policemen who stopped a major crush at one side of the Holgate End.

Burnley got to the final and lost a dreary match to Charlton - and just as in 1913 got promoted. Boro, just as in 1913, responded well, with a draw against leaders Wolves, and then went into Typical Boro mode again with just four points from their last 12 games and finished 10th.

The League Cup:

Burnley 0 Middlesbrough 2, League Cup fifth round, December 1975. Michael Holding was playing his first test, and Ronnie Wood had just joined the Rolling Stones.

The omens weren't good - in the league game earlier in the season David Armstrong put Boro ahead after five minutes - then Peter Noble hit a hat-trick as they let in four. Typical Boro again, crushed by a Burnley team who were eventually relegated.

This time the crowd was only 15,509, as Burnley were third-bottom of the First Division and had just lost five in a row. Boro were ninth, with one in in six games, having just lost at champions Derby, who they had beaten in the previous round.
David Mills scored after two minutes, had the Manchester papers comparing him to Denis Law, and Willie Maddren added a second just before the end. At Turf Moor, of all places, where they had only won twice, Boro got to a major cup semi-final for the very first time.     

The Aftermath: Burnley recovered for a draw against eventual champions Liverpool three days later, but they spent all but a week in the bottom three and went down for 33 years.
Despite that the Boro win also helped their manager Jimmy Adamson out of his job, with his fate sealed when they lost to Blackpool in the FA Cup two weeks later. 
Boro responded by signing Phil Boersma, drawing with Manchester United then winning the Anglo-Scottish Cup at Fulham a week later. The semi-final against Manchester City looked even more historic when John Hickton scored the only goal in the first leg at Ayresome Park, but then came another organisational disaster in, or rather before, the 4-0 defeat in the second leg, with the club refusing to pay for an overnight stay, gridlocked traffic, with the team arriving half an hour before kick-off and many of the fans after it, and, Willie Maddren's particular pet hate, just one pair of pliers to change several dozen studs.

Burnley 3 Middlesbrough 2;  Middlesbrough 1 Burnley 1, League Cup second round, October 1982. Cats had just opened on Broadway, the Mary Rose raised, and Princess Diana was showing baby Wills off.

Once again Boro had already lost 4-1 in the league to Burnley, this time at Ayresome Park, the first home game after relegation in, and only 8,000 turned up. It was also Typical Boro again, as Burnley didn't win another away game for 16 months and were relegated.
Burnley were fourth-bottom and had lost their last four games. Boro were rock bottom and had just sacked manager Bobby Murdoch after two more 4-1 home defeats, and Harold Shepherdson was caretaker while talks were going on in (cigar) smoke-filled rooms with Malcolm Allison.

But Boro took the lead through Heine Otto just before half-time, but not quick enough before it as Billy Hamilton equalised in injury time. Terry Cochrane put Boro back in front in his first game back at his old club, only for Mike Phelan and Kevin Young to give Burnley the win.
The second leg was Allison's first game in charge, and it was going well in front of the first five-figure crowd of the season, with Ray Hankin, another ex-Burnley player,  scoring his first goal after only seven minutes. But the man who ruined everything with an equaliser in the second half was one Brian Laws. And he'd also scored in that 4-1 win. But he did redeem himself a few times in the following seasons.

The Aftermath: Boro went on to complete a six-game unbeaten run - something they wouldn't better for four years - but only survived with a draw on the final day to finish 15th. The team they drew with also narrowly survived - Chelsea.
Burnley went on to beat first Division sides Coventry, Birmingham and Spurs and reach the semi-finals, where ether lost to Liverpool. And they got relegated.