At home, they find little comfort, or victories.
Away, they can't stop scoring goals, and winning, as they did again by crushing Hearts 4-0 in Edinburgh last night.
The reason for this split personality baffles manager Neil Lennon. Perhaps it's the spikiness of the atmosphere at places like Tynescastle. Maybe it has got something to do with the fact opposition teams are more willing – and compelled – to come out of their shell and have a go.
It could be a combination of both. Whatever, it becomes a potent mixture which cures all that may be ailing the SPL champions in the defence of their title.
While the rest of the pack engage in notions of topping the table, Celtic are waiting to be overtaken.
Sure, their progress towards another championship has stuttered rather than purred. Dropping 17 points has made this the worst start to a league campaign since 1998.
However, there is no doubt the Green Machine possesses another gear – at least one more – which can be engaged when the need arises.
When they get going, the rest labour in their slipstream, and it is to their credit that so many remain within touching distance as we approach the halfway point of this wierdly-compelling season.
The fear they must all harbour is that, when Celtic do get it all together on a consistent basis – as Lennon believes they will once they get their final Champions League Group G tie out of the way on Wednesday – the Hoops will disappear over the horizon at a rate of knots.
Certainly, the manner in which they destroyed Hearts sent out a message, if not a warning, of precisely what they are capable of producing.
The memory of the bitterly disappointing showing against Inverness at the weekend was swept away – along with Hearts' hopes – amidst a first half-hour salvo.
Lassad Nouioui led the way, volleying home Kris Commons' cross with only 10 minutes on the clock.
Mikael Lustig doubled their advantage 11 minutes later when he was rewarded for clever build-up play and dogged perseverance, collecting a blocked effort from Commons and rifling low into the net.
When the hapless Ryan Stevenson directed a Charlie Mulgrew cross beyond his own keeper on 29 minutes, the job was done, as were Hearts.
The ubiquitous crackling atmosphere which makes these fixtures special had much of its cordite removed by Celtic's rocketing start.
The same could not be said for some misguided Hoops fans who had managed to smuggle a flare and some bangers into the stand they occupied.
Gary Hooper put the finishing touch to the night with the fourth goal five minutes from time, and Lennon will be delighted that his main striker is now back on target.
The Celtic boss can now look forward to this weekend's William Hill Scottish Cup-tie against Arbroath assured that the blue touch paper has been re-lit in his players, and that they are ready to try their best to explode into the last 16 of the Champions League when they face Spartak Moscow next week.
It will be a much-changed side which faces the Russians, with the need to rest some of the walking wounded very much in Lennon's mind.
Others, like Scott Brown – who again started on the bench last night before replacing Joe Ledley for the last quarter of the game – might need a run-out to get them back up to speed.
Having left some of those who misfired against Caley Thistle – including Tony Watt and Georgios Samaras – out of his starting line-up, the chances are there will be a few knocking on Lennon's door asking to be given the chance against Arbroath to remind him what they have to offer.
No-one will want to miss out when the team to face Spartak is announced, and retaining this hunger for all matches is one of the biggest challenges facing the Hoops boss.
Lennon can take solace from the fact that, no matter the game, the opponent or the occasion, there remain some constants.
Fraser Forster's form is one, and the win over Hearts would not have been nearly as comfortable or comprehensive had the keeper not pulled off a reflex save from a point-blank Marius Zaliukas header in the first half, then an even better stop from Kevin McHattie's free-kick just after the break.
Forster actually saved his best for last, diving full length to stop a Zaliukas penalty in the dying minutes after Kelvin Wilson was adjudged to have brought down Stevenson in the area.
The sight of the consistent Ledley back in the side and showing no sign of being impeded by his long-standing groin injury will also offer some comfort to Lennon.
Slowly, but surely, he is getting some of his big players – like Beram Kayal – back on the pitch, and others – like Wilson – back into form.
With Emilio Izaguirre set to join them this weekend, all that's left is for James Forrest to recover from the hamstring injury which has kept him sidelined for the past month.
Oh, and to see Celtic's away league form find its way to Parkhead.