That has been the safety valve at particularly fraught moments in one of sport's more stressful jobs. 'He has been known to clear the top of his desk with a motion like a windscreen wiper,' says a colleague.
At the end of this month, he will clear his desk in more restrained and conventional style and cease to be the game's chief executive.
The Oxley years, which are ending with compulsory retirement at 55, have been a remarkable time for rugby league, but hardly more remarkable than the manner of his arrival.
Although he had grown up in Hull and retained a keen interest in the game of his youth, Oxley in 1974 was about as far away from the roots of the code as it is possible to be, without falling into the Channel.
He was headmaster of a minor public school in Dover. Young and ambitious, the next move would have been to one of those institutions upon whose playing fields the battles of empire were allegedly won.
'I had never lost touch with the game,' he recalled. 'The lads always knew that if they didn't fancy going through the niceties of Macbeth's soliloquy they could get me talking about rugby league instead.'
You had to be a true believer to wax enthusiastic about the game to an audience of public schoolboys in Dover in 1974.
'Gates were down and the game wasn't being played in the right way. People have said the game was dying; it wouldn't have died - there were too many good people involved to let it die - but it was in trouble.'
Against that background, the power-brokers of the game decided that the 29-year reign of its secretary, Bill Fallowfield, had to end.
'It says something about the horizons of the game in those days that the advert for the job was at the bottom of the Situations Vacant in the Daily Telegraph and was about this big' (thumb and forefinger very close together). 'More out of curiosity than anything, I applied.'
When Oxley was interviewed by a daunting cabal of nine former chairmen of the League, the decision had already been made to give the job to Bev Risman, a dual union and league international who stood out among the field of candidates. Like a man who knows he has already failed his driving test, Oxley went in with the attitude that he had nothing to lose, and forced the chairmen to change their collective mind.
With his energetic PR man, David Howes - 'We were the two Davids long before the two Ronnies' - Oxley set about raising morale and changing both the style and substance of rugby league.
His own style came as a bit of a shock in the heartlands. Articulate, almost donnish, his turtle-neck sweaters and a grin, which, like the Cheshire Cat's, seemed to have an independent life of its own, became unlikely trademarks.
The sweaters were phased out; the grin remains, although slightly worn around the edges by 18 years as the public face of professional rugby league.
Plenty has happened. The image of the game has improved immeasurably, gates have reached a plateau upon which they have done well to remain during such a long recession; standards of play have undeniably risen. 'We are so lucky with our players,' said Oxley, suddenly the star-struck young man from Humberside again. 'One thing I hope as I leave is that they realise the esteem in which I hold them.'
Few sports have courted sponsors as energetically or as successfully. The administration of the game is sleeker; its once seedy offices are now a showpiece. Upheavals like the introduction of players' contracts and ground safety have had to be negotiated.
And yet much remains to be done by his successor, the former Wigan chairman, Maurice Lindsay, whose appointment - the game's worst-kept secret - will be announced today.
More clubs are in financial trouble than ever, and there are no crocodile tears from Oxley when he says that it is 'almost inevitable' that some will fold.
The professional and amateur wings of the game are at war, although Oxley, interestingly, thinks that strife will prove neither as long-lasting nor as damaging as many fear.
The domestic and international expansion of the game rests firmly among the unfinished business, and the declared ambition to be 'truly national' in time for its centenary in 1995 will not be attained.
That disappoints and angers the game's zealots, who lay some of the blame at Oxley's door, alleging that his activities with the Sports Council and other bodies - which will be important elements in a busy 'retirement' - have made him remote from the sport he has been employed to serve.
It is a charge he refutes. 'If you're not at your desk, people assume you're on holiday. In fact, I've forgotten what a holiday is and rugby league has benefitted from my wider involvement.'
There are unrealistic expectations within the code, he believes, which inevitably lead to disappointment.
'People in the game claim very readily that it is The Greatest Game of All, and of course we believe that it is. But then they think that if it is not sweeping the country, it must be because it isn't being marketed properly. That ignores the fact that there are 15 million people who prefer soccer and always will, and four million who prefer rugby union and always will.
'If you can hold on to your market share, you aren't doing badly. If I have had one frustration, it's something which I, as a Yorkshireman, have come to think of as a Yorkshire characteristic.
'While other sports trumpet their often very flimsy wares from the rooftops, there's an element in rugby league which is never happy unless something's going wrong.'
The Oxley years have afforded the moaners some pleasure, in that case, but they have also been an era of dramatic progress on many fronts.
If rugby league in south Cumbria was in a better state, David Oxley's obituary could read that Harrow's loss was Barrow's gain.
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