On the one hand, Cook, the England captain, is a naturally cautious leader and a functional tactician.
On the other, Bayliss, the England head coach, preaches positive, attacking cricket – a message that he felt the need to personally reinforce ahead of today’s fourth Test against India in Mumbai in a departure from Cook performing the team talks.
As England sought to hit back from 2-0 down, a seemingly fanciful idea on the evidence of their winter so far, Bayliss spoke of his desire to take a more prominent role in the dressing room and to encourage the side to play more positively.
“It will be up to me to play a role there,” he said of the side’s preparation for the game at Wankhede Stadium. “Probably over the last few series, Cooky has taken on more of a role there, which has been great.
“But I think for the coming couple of Tests it is time I stepped up to the mark to just remind them of how we have played when we have played well.”
Bayliss felt that England’s mindset was “more along the lines of survival” as they slipped to defeat in Mohali and Visakhapatnam, following a good performance in the drawn first Test in Rajkot.
Although he was not publicly criticising Cook, more emphasising his own accountability for results and performances, he is clearly concerned with what he has been seeing.
In an interview last summer, the 53-year-old emphasised that “the boss of the cricket team is the captain” and that “my job is to try to get the captain to take on more responsibility and make the players become more self-reliant”.
Bayliss added: “They are the ones who have to make decisions at the right times out in the middle, and if someone off the field is telling them what they should be doing every minute, then they don’t learn to make a decision on the spot.”
However, Bayliss, hitherto happy to take a back seat, has now grabbed hold of the steering wheel in an effort to avert what could yet degenerate into an embarrassing hammering against the world’s best team.
Indeed, on the evidence of England’s recent displays, the suspicion remains that Cook and Bayliss – although excellent men in their own right – are not exactly soul-mates in a cricketing sense.
In many ways, Cook is quintessentially English and Bayliss quintessentially Australian, give or take the odd characteristic.
Bayliss greatly admires flair players such as Ben Stokes, saying that “some of us are in awe of what he does”.
Cook could well be in awe of him also, but it is difficult to think of two more contrasting players than Cook and Stokes, with Stokes seemingly more of “a Bayliss man”.
The cricketing compatibility of Cook and Bayliss is also pertinent given that Cook cast doubt on his own captaincy future going into the series.
“Deep down, I don’t know how much longer I am going to carry on,” he admitted. “It could be two months, it could be a year.”
Some say that Cook should stand down if England lose the current series, but Bayliss said this week that he is confident that Cook will remain as Test captain for next winter’s Ashes in Australia, after which Cook may indeed resign – win, lose or draw.
As such, a parting of the ways may or may not be imminent, but it would perhaps be no bad thing if Cook focused solely on what he has traditionally done best – score runs.
Of late, England appear to have stagnated under his leadership, which has been under the microscope too often for comfort.
Since winning in South Africa last winter, England have won five Tests and lost five, failing to sparkle last summer against Sri Lanka and Pakistan.
Too often, England have been inconsistent – up one minute and down the next.
Despite a number of talented players, not least in the all-rounder department, they have struggled to translate those talents into results.
Joe Root is widely expected to be the next England captain, with Ben Stokes a good bet as his deputy.
A Root/Stokes partnership would seemingly be more likely to guarantee the type of direction that the coach clearly wants.
England are at their best when they play in a positive fashion, for they have the players ideally suited to that style.
For Bayliss to fulfil his potential as England coach, you sense that he would benefit from working alongside a captain who is more in tune with his own way of thinking.