My general idea for the Coruscant Guard is they're often the very first clones that Jedi - especially older initiates - meet. Sometimes by chance. Often by design. Any Jedi heading out first works with the CG to get an idea of what it's like to work with the clones. It's a programme both groups created and agreed upon.
More importantly, older initiates and Padawans of all ages are sent to help fill out forms and learn basic GAR procedures before heading out.
And Corries in turn are welcomed to the Temple as they please. It's common to find them in the Archives or teaching combat. Or surrounded by a gaggle of younglings who insist their faces might be the same, but they look so different in the Force it's impossible to get them mixed up.
Those who want are welcomed to help crèche masters handle their little ones. Encouraged even. Children cause enough chaos, but Jedi children? Even more so. This trust isn't something the Corries take lightly; they know they're being given something no person outside the Jedi ever has. It's a sign of deep trust and something that gets them even more respect from their brothers elsewhere.
Which means the CG get to know many young Jedi very well. Newly chosen Padawans will often go to their favourite Corries and tell them who their new Masters and Legions are.
The CG are very protective over the little Jedi. They keep tabs on how they're doing, giving Padawans personal comms to contact them if there's ever a problem with their men. Keep tabs on their Masters. If the Jedi are the Clones' voice to the Republic, the Corries are the Jedi's voice to the other clones. They know the Jedi as a whole the best and place peacemakers when needed.
They best know the deep strain the Order is under. The fine line they constantly walk, and how often they're under threat. They have strategically placed watchmen keeping an eye on the routes Jedi most often use.
And because they're Corries, the men automatically hold them in much higher regard, sometimes even above their own Captains/ Commanders, so they'll listen.
The Jedi also have a rota of healers working shifts in the CG medbay, sharing medicines, techniques, and helping restock supplies because going through official processes gets... difficult. It took some time to get the men comfortable around them, but they were gently persistent and it did wonders in keeping them healthy. Ofc, the Temple Healing Halls are always open for the men, but not every trooper can be transported, and sometimes it's safer not to advertise this to the public.
And in general, the men are always encouraged to come and ask questions. To learn. Rest and recover somewhere there aren't cameras on them. Inform the Jedi about any problems they're facing eg. Anti-cloners and the like threatening them. Not because they can't handle it themselves, but because the Jedi can mitigate further damage to the clones' reputation should it escalate to violence.
The men are trained to act as riot police. The Jedi encourage restraint and more importantly, explain why. Seeing how it keeps peace where things would become very messy, this mindset is swiftly adopted by the Corries, with a prepared list of reasons why they acted as such should a Senator come asking.
Then there's other things like off duty Jedi being on call should the CG require additional help in investigations. There's a quick and simple process to ask for Jedi assistance.
The Coruscant Guard don't have a Jedi.
They have the whole Order.
(I also like to think the Temple Guard and Coruscant Guard have a close relationship in their own way. They communicate primarily with subtle but complex body language.)
Also expanding on the Coruscant Guard compared to other clones + perception. The conditions are definitely harder in terms of mental strain. But I feel like even if they were originally given run down barracks and the like, between their own ingenuity and later having access to the Jedi's supply of stuff, it's actually a really comfy place to wind down.
The other clones deeply respect the Corries. There's a gritty reality to life on Coruscant where you're expected to remain a nameless guard to the elite, and aren't particularly liked by the public. It's understood as a difficult position even without the addition of being under constant scrutiny with no room for error.
Venators are 95% clone and most Jedi outrank the other natborn officers, so easily deal with the occasional problem/threat. Coruscant very much isn't. The moment you step out the barracks, you're in the spotlight. If you're caught, it's straight to a military court, and though they try their hardest, there's only so much the Order can actually do (hence why they encourage studying law and the like.)
This culminates in a very genuine respect rather than something to toss around and joke about. Any clone caught making light of the CG is given a very harsh reality check by their other brothers.
Because honestly? Coruscant makes Kamino look fun.
It's a different kind of battlefield. One most of the men know they aren't cut out for.
