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As the title says, I am unsure as to whether a creature with a climbing speed would be able to climb across the ceilings using their climb speed and avoid any difficult terrain penalties.

This question came about because of a feature of a Simic Hybrid from the Guildmasters' Guide to Ravnica, which gives a player character a climbing speed.

A creature with a climb speed can climb across ceilings but not any better than a creature without a climb speed.

Per RAW, creatures with a climb speed only gain a benefit to climbing vertical surfaces as outlined in the MM.

Climb

A creature with a climbing speed can use all or part of its movement
to move on vertical surfaces (emphasis mine). The monster doesn't need to spend extra movement to climb. (MM p. 8)

Creatures with climb speeds do not explicitly gain any benefit to moving across ceilings, though any creature that could reasonably scale a ceiling may still do so. A creature spends 2 feet per 1 foot of its movement scaling a ceiling regardless of whether or not it has a climb speed.

If any character or creature has themselves, the capacity to scale the underside of a horizontal surface (i.e. ceiling), and if that surface has enough outcroppings or other characteristic that would make such a feat possible, then that creature can do so using the climb rules outlined in the PHB.

Climbing, Swimming, and Crawling

When climbing ..., each foot of movement costs 1 extra foot (2 in
difficult terrain, unless the creature has a climbing ... speed. (PHB p. 182)

It should also be noted that having a climbing speed does not necessarily exempt you from having to make athletics checks under duress or when failing a climb could have consequences. The following rules, per RAW, still apply.

Ability Checks

... The DM calls for an ability check when a character or monster
attempts an action ... that has a chance of failure. (PHB p. 171)

and

Using Each Ability

Strength Checks

Athletics. Your Strength (Athletics) check covers difficult situations you encounter while climbing... Examples include...
climbing a sheer or slippery cliff [ceiling], avoid hazards while
scaling, cling to a surface while something is trying to knock you
off. (PHB p. 175)

I'm not sure precisely what you're asking here.

Can you hang from or cross the ceiling?

Any creature, climb speed or not, can potentially grab onto a ceiling feature and hang from it. If the ceiling is covered in vines, carvings, rafters, or has some other potential handholds, it's up to the DM to determine whether the handholds can support your weight, and what roll (if any) is needed to maintain a grip and/or move across the surface hand-over-hand.

While most of the climbing rules talk about 'vertical' surfaces, it's pretty clear from our normal understanding of the world that an action like this would still constitute climbing and be subject to all the rules that relate to that, such as increased movement cost and requiring an Athletics roll if the surface is challenging. A climb speed operates as usual in this case, allowing you to ignore the cost increase and potentially not make rolls.

(As examples, I would think that crossing from something like playground 'monkey bars' would not require a check, or only a minimal DC, while more difficult surfaces, such as a vine-covered ceiling or going from stalactite to stalactite, would be far more difficult.)

Can you hang from or climb across a smooth ceiling that lacks handholds or other similar features?

Probably not, unless the source of your climb speed (or some other rule) specifically says you can.

For example, the spider climb spell says

one willing creature you touch gains the ability to move up, down, and across vertical surfaces and upside down along ceilings

Similarly, the spiders in the Monster Manual share an ability called (again) Spider Climb, which reads:

The spider can climb difficult surfaces, including upside down on ceilings, without needing to make an ability check.

A Simic Hybrid's climb speed (if I recall correctly) is just a climb speed that doesn't include a spider-climb-like clause, so it would not allow you to cross a ceiling without something to hold onto.