Yet Cheika has made a point of avoiding any form of verbal confrontation with England this week.
'I think he understands the psychology of the game pretty well,' said Jones. 'I think he understands England are under pressure and he doesn't need to do anything to add to that. And he doesn't want to give them anything to feed off. Check's a smart, shrewd operator.'
Jones says he was always cut out to be a coach. 'He wasn't the captaincy type but he was still the sort of bloke people followed,' he said. 'And, to be honest, great captains don't always make great coaches.'
Cheika is a fine coach. He guided Leinster to the first of their three Heineken Cup triumphs in 2009 and led the New South Wales Waratahs to their first Super Rugby title last year with an exciting brand of running rugby. He became the first coach to win the premier club competitions in both hemispheres.
Brian O'Driscoll was certainly impressed by Cheika during their time together at Leinster.
'He changed the mentality of many, many players,' O'Driscoll has said. 'He provided the foundations we were able to build on.'
Cheika is regarded as an innovator who has injected ambition as well as some extra steel in a Wallabies side he took charge of only last autumn after the shock resignation of Ewen McKenzie.
He was appointed despite being regarded as a bit of an anti-establishment figure; a maverick who has had his brushes with authority.
When he was in charge of the Waratahs he received a suspended ban for a confrontation with a South African sideline cameraman during a defeat by the Durban-based Sharks. Another time he slammed the door to a coaching area so hard he smashed a window.
"Sometimes you've got to show the players you care," Cheika said in defending his tantrums.
He is undoubtedly bright, however. A playing career that took him to France and then Italy enabled him to learn languages and he is now fluent in French and Spanish as well as Arabic, courtesy of his Lebanese parents.
On Sunday, after overseeing a comfortable victory against Uruguay, he conducted interviews in a variety of languages. The 48-year-old married father of four was charm personified.
Such social skills may partly explain his success in business, even if the fact that he has made millions in the women's fashion industry just adds to the perception of him as a combustible, cultured contradiction of a man who sports matching cauliflower ears and a long scar that is visible under his hairline; an injury sustained when he was 'scalped' by an opponent's boot.
His career in industry took off when he worked as a business manager for the leading Australian designer Collette Dinnigan. It developed further when he set up his own distribution company that secured the rights to a jeans brand made famous by Victoria Beckham.
Dinnigan recently told the Sydney Morning Herald: "He didn't tolerate any bull**** and would be firm with business associates that were not delivering, yet he would very elegantly greet all our international fashion buyers in Paris and address them in various languages."
Some think Cheika is driven, like many top coaches, by having failed to fulfil his ambitions as a player.
He was unable to emulate many of his Randwick colleagues in representing New South Wales and the Wallabies, although by his own admission he simply 'wasn't good enough'.
Others, however, believe it runs deeper than that and comes from always having been an outsider. He was the son of a Lebanese immigrant who was said to have arrived in Australia with little more than 10 dollars in his pocket.
He was not educated at what would be considered a typical rugby school. He was not a member of Sydney's elite, growing up in the more working-class coastal district of Coogee.
Jones, the son of a Japanese mother, agreed with that theory. "You've hit the nail on the head there," he said. "One hundred per cent.
"That Randwick dressing room was a real microcosm of Australian society. A great bunch of blokes who all got on really well.
"But you've got to remember this was Australia 30-odd years ago. It was still a very white Australia and rugby was very much a white-collar sport, so if you were like Check and myself you really had to prove your worth.
"Back then there was a bit of racial sledging. Both Check and I had to put up with a bit of that when we played and you had to be able to fire back at opponents on the field.
"It's probably a reason why we always got on well. Probably because we weren't quite part of the establishment. We shared a lot of interests. We both loved rugby league."
Some describe Cheika as a control freak. 'He takes over everything,' said another observer.
"He becomes coach, CEO, director of communications. At the Waratahs the CEO was completely sidelined because Check just ignored him."
Jones, however, likes his style. 'After I finished as Australia coach I was probably a bit resentful for a couple of years after,' he says. 'But when Check took over he changed my whole view of the team again. I really like what he's doing and I really want him to do well.
"A lot of coaches these days seem to be speaking from a management manual. Check wears his heart on his sleeve and he's good for the game. And I like what he's doing with the Wallabies. They play bold, aggressive rugby. But he's also made them tough mate, really tough."
