McNeill, who captained Celtic when they beat Internazionale to become the first British winners of club football's greatest prize, takes up a story which somehow encapsulates the surreal nature of a sweltering afternoon 30 years ago yesterday.
"Several of the boys wore false teeth, and our keeper, Ronnie Simpson, used to keep them in his bunnet [cap] and leave them in the back of the net. We were back in the dressing-room with the cup when Bobby Lennox suddenly said: 'Where's the teeth?'
"The funny thing is that they were still lying there, even after the best part of 12,000 Celtic fans had poured on to the park. They'd grabbed whatever souvenirs they could but overlooked this old cap. Otherwise there would be some middle-aged Scot today boasting that he'd got the Lions' teeth!"
Of that there can be no doubt. Ninety minutes of football, fought out long ago in a distant arena, still exudes a powerful mystique for many Scots. Not having been born when McNeill raised the giant trophy is no bar to reliving the glory.
Now a grandfather of 57, the centre-half known as "Caesar" would have had trouble lifting his medal after undergoing triple heart by-pass surgery in February. However, McNeill has recovered so well that he recently accompanied a television crew on a nostalgic return to the Estadio Nacional. Next month he will lead the historic team as guests of the North American Celtic Supporters' Association at a convention for 1,800 in Las Vegas.
At a dinner to mark the 20th anniversary, McNeill received such an ovation that the board felt compelled to bring him back for a second spell as manager of the club with whom he won a staggering 23 medals. So it is surprising that he is genuinely taken aback by the latest wave of interest in Celtic's annus mirabilis.
"I don't think you'll ever get a situation like that again," McNeill reflects. "We won the European Cup with what was effectively a Glasgow district XI. Apart from Bobby Lennox, who's from Saltcoats, 30 miles away, we were all from within 15 miles of Celtic Park. Nowadays, with freedom of contract, we'd have been picked off.
"Scottish football was good at the time. Our clubs regularly went long distances in Europe; not just Celtic and Rangers, but Kilmarnock, Dundee and Dunfermline, too. We've lost our way since then, perhaps because money is too easily come by in a small country like ours and because we don't work hard enough at our game.
"We've lost aspects of our football that were attractive and successful. We had pride and passion, but also great creative players like Bobby Murdoch and Bertie Auld plus skilful dribblers like Jimmy Johnstone, and not just at Celtic."
For all the popular perception of Stein's Celtic as a swarm of free spirits, their most enduring symbol is the rock-like McNeill. His towering header won the first trophy of the era, the Scottish Cup in 1965. A similar effort saw off Yugoslavia's Vojvodina Novi Sad in the final seconds of the quarter- final, one of two moments that season which persuaded him that destiny was taking a hand.
"The other was when Stevie Chalmers got the winner in Lisbon. Joe McBride was our top scorer but he'd picked up a bad knee injury at Aberdeen. That was the only reason Stevie was playing, which always makes me wonder about fate."
Despite a clean sweep of domestic silverware, Celtic flew to the Portuguese capital as undisputed underdogs. The Champions' Cup was the preserve of the Latin nations. Some of the European media portrayed the Scots as innocents abroad, out for a few days' sunbathing.
Stein, a stickler for discipline, knew better. "We had a beautiful hotel with a fantastic swimming pool, but Jock believed the heat would tire us out so he limited the time we spent out there," McNeill explains. "He kept stressing that we weren't on holiday."
The regime did allow some unconventional behaviour. The night before the final, McNeill and his colleagues took up an invitation from a Scottish expatriate to walk to his villa and watch England play Spain on TV.
"Neilly Mochan [the trainer] was with us and he was legendary for getting lost. We ended up wandering through woods and clambering over rocks in the dark. With hindsight it was crazy, though it took our minds off the match and eased the tension."
In fact it was Inter, over-dependent on cynical defence, who were probably under greater pressure. If Celtic had lost, McNeill is convinced people would have applauded them for simply reaching the final.
The main thrust of Stein's team talk was to enjoy themselves. "I know it's a cliche, but it was right for us. He also used the fact that Inter had reneged on an agreement about who'd train first in the stadium. The chip on the shoulder works for us Scots."
The image of the teams waiting in the tunnel compounds the retrospective incongruity of it all for McNeill. "The contrast was incredible. They had all these lovely, lyrical names like Alessandro Mazzola, Giacinto Facchetti. We had Murdoch and Auld. They had tanned legs and faces, like models. We were all gums and pale skin.
"Then Bertie started us all singing: 'It's a grand old team to play for'. The Italians were bemused, possibly even a little intimidated."
Although Inter scored from the spot in the opening minutes, it was they who proved toothless. Their disinclination to try to add to the lead enabled the hooped shirts to build up an irresistible momentum.
"They thought one goal would do it," McNeill remembers. "They didn't appreciate the quality of our players, so they allowed us a lot of possession. We probed and probed - it was a siege - but we kept hitting the woodwork and had a blatant penalty turned down."
At half-time, Stein had instructed Celtic to forget any sense of injustice over Inter's goal. McNeill can still hear him urging: "Just get out and play." Ironically, the goalkeeper identified by Stein as a weak link, Giuliano Sarti, was in inspired form.
Sarti had as much chance of stopping Tommy Gemmell's brutally struck equaliser, midway through the second half, as of catching a bullet. And he was helpless when, with five minutes left, Chalmers diverted Murdoch's drive: 2-1.
"I steeled myself for an onslaught, but it never came. Inter were knackered because they'd had to do so much covering and chasing. It was 85 degrees but we were extremely fit and adept at keeping the ball."
While the scoreline scarcely reflected Celtic's superiority, the manner of their success enchanted a Continent. A French reporter dubbed them "L'Orage" - the storm - while a Swiss commentator hailed their "alchemy of will and technical brilliance". An alliterative Scottish scribe labelled them the Lisbon Lions, presumably grateful that the final had not gone to Zagreb.
The last blast of the whistle had come with Stein on walkabout, unable to look. Soon he was engulfed by delirious fans. "There was a six-foot moat," McNeill recalls, "but, with that kind of elation, a wee bit of water wasn't going to hold them back."
After retreating to their changing room, where Liverpool's Bill Shankly told Stein he was now "immortal", Celtic went out to collect their trophy and their teeth. The green party was only just beginning.
Glasgow belonged to Celtic. The bus came on to the tarmac at the airport and crawled to Celtic Park through crowded streets. There were 65,000 inside as the squad, led by an Irish accordion band, took the cheers from the back of a lorry. One man scaled a floodlight pylon for a better view.
When McNeill finally wound down, he saw the pictures of a deserted city centre at kick-off time. It tickles him to think even the Rangers supporters must have been watching.
If the celebrations are tinged with sadness for Caesar it is because they find Celtic riven by internal strife in the forlorn pursuit of Rangers; and because Stein and Mochan are not around to join in.
The players do not see each other often - one, Willie Wallace, lives in Australia - but there is still a "wonderful bond" when they do. "Stevie Chalmers and myself are the only ones who've had health problems, probably the last two you'd have expected." Happily, McNeill has already resumed his role as a television summariser.
Talking of TV, a grainy old clip which featured in Hugh McIlvanney's homage to Stein, Shankly and Busby stirred memories of that heady spring for McNeill. It showed Stein and his youthful charges receiving the Team of the Year trophy from the BBC's Sportsview programme.
"Though it was a British award, we tended to consider it an English thing, and Jock made a point of saying we'd won it for Scotland. I'd forgotten that Matt [Busby] made the presentation, and that Jock said he hoped Manchester United went on to win the European Cup themselves. Which they did, but Celtic were there first."Reuse content