Yet within three years the relationship had soured. Mr Taylor, the story goes, made the fatal mistake of disagreeing with Sir Paul on too many issues and found himself a progressively isolated figure.
Eventually he found the boardroom was being rearranged around him. Ernest Mario, then head of Glaxo's American operations, was made overall chief executive. Overall meant over Mr Taylor. Not surprisingly, he resigned.
Last week it was Dr Mario's turn. 'Differences of opinion over the running of the business,' said Dr Mario, as he handed in his resignation.
That was not how Glaxo put it. A 'new and strengthened structure', said the release detailing new board arrangements. The resignation was saved until the fifth paragraph.
Like most international companies in the pharmaceuticals sector, Glaxo is notorious for its secrecy. Sir Paul himself has a Delphic way with words at the best of times. At the worst of times, he says things that are downright contradictory.
In person, he prefers to talk obliquely about problems with 'the chemistry of the two people at the top' when discussing Dr Mario's resignation.
But few doubt that there was a significant disagreement between the two men about future strategy. Even Sir Paul, when pressed, makes it clear that as far as he was concerned, there was no dispute because he could have overruled Dr Mario as and when necessary.
The differences seem to have stemmed from Dr Mario's view that Glaxo should follow an aggressive, acquisitive, expansionist approach.
Dr Mario, it seems, favoured Glaxo selling its drugs over the counter, possibly through the purchase of a big OTC drug company, which would have needed to be financed by an equally large rights issue. Many in the drug industry believe OTC sales will substitute for profits lost if governments restrict the number and cost of prescription drugs.
But for Sir Paul this was worrying territory. He had forged Glaxo's reputation from extensive research and development and the production of high- price, high-quality, heavily marketed prescription drugs.
Such a risky switch of emphasis made him nervous and when Glaxo started to attract criticism for what looked like dithering and dissent, he clearly felt something had to give.
It took time, though - more than a year. As Sir Paul, with characteristic opaqueness, describes it: 'The tensions began to grow at the beginning of last year. On that sort of scale. I'm sure he became a little less happy. What happens is you begin to isolate yourself. Subconsciously. There was a growing unease about that. But he was still supported, because you have to take off your hat to a man of Ernie's stature.'
By last Wednesday, Dr Mario had found that the board had been reorganised in such a way that his own power was diminished. It was something over which he had no control. As Sir Paul says: 'He would have had to accept it.'
Dr Mario, cushioned by a pay-off likely to approach pounds 3m, did just that and Richard Sykes, head of R&D, inherited the chalice. Intriguingly, the scientifically minded Dr Sykes has been given a commercially minded deputy - Franz Humer, the chief operating officer - in what seems a partial splitting of the chief executive's role. Dr Humer, for instance, gets Europe to run; Dr Sykes the United States.
Equally, two directors - Jeremy Strachan, responsible for corporate affairs, and John Hignett, who looks after Glaxo's pounds 1.5bn cash pile, will report directly to Sir Paul.
To Glaxo, it may be a question of strengthening the board; to outsiders, it looks ike autocratic divide-and-rule.
Not that Sir Paul is against absolute power. 'I don't think you would get much support for talk of autocracy here, but then what's wrong with autocracy if it works?'
The reshuffle also leaves Sir Paul, who still insists he will go in three years' time when he is 70, without a clear successor. Yet ask about this and he responds: 'Is that bad?'
He insists 'the most likely thing' is for a 'move up of competent people', with Dr Sykes the most probable candidate. But he leaves the door open. 'It could be that if Richard didn't want to or whatever, another candidate could jump over.'
He views the uncertainty as a reflection of 'robustness' in Glaxo's top management. 'When that opportunity was there four years ago, we didn't have anyone. We didn't have anyone even to support Ernie.'
Seemingly oblivious of the implied criticism of Dr Mario, he muses: 'Looking back, we made whatever we could do. And Ernie was not in those circumstances a mistake. But the situation has changed. More strength has come in. And we have a much stronger, really a much stronger organisation than we did.'
So far, he has failed to convince the City. Glaxo's battered shares lost yet another 19p on news of Dr Mario's departure.
As one analyst observed: 'To lose one chief executive may be a misfortune; to lose two looks like carelessness.'
But even bigger issues dog Glaxo: above all, how it will replace Zantac, its anti-ulcer drug. Zantac accounts for more than half Glaxo's profits, but the patent on the first version expires in 1995, while that on the currently prescribed second version ends in 2002.
Sir Paul may have decided on Glaxo's medicine: no more talk about OTC solutions, mega-acquisitions or rights issues. It is back to the prescription drugs formula that has produced past success. But as yet, there is no evidence the remedy will work.
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