tl;dr: It is up to the GM to decide which up is relevant and hence if you can have your dome upside down.
Probably, when writing the spell description, nobody thought about the possibility of casting it upside down, i.e. it was assumed that the caster's up is identical to the gravitational up, which it usually is. Now, the spell description says "dome" indicating that you cannot gain a sphere all around.
If you have created a situation where this is no longer the case the GM needs to decide which up is relevant. There is nothing in the spell description that would suggest that you should not profit from it while upside down. You cannot profit from the spell if you are upside down and the spell considers the gravitational up for "above you". It seems therefore entirely reasonable to allow casting the hut upside down if there is already a situation where this would be relevant.
When playing an RTS like Age of Empires or Supreme Commander, you cannot position a building on a slope. That is not how it's like in reality. Probably you have seen a mountainside village before on vacation or otherwise where this works just fine. Also, above and around does not exclude below. If the dome were to be positioned on a slope with ground contact, it is still above and around even if it is partially below as well. D&D is not, and is not intended to be, a perfect simulation of reality, but in this case we should assume the reality variant and not the RTS variant. The spell does not detail where it can be placed but it seems reasonable to assume that it will adapt to the grounds and completely enclose the PCs so that no snakes and zombies can crawl under it even if the ground is not perfectly flat.
Tiny Hut creates a dome. The dome is a result of the area of effect rules.
Tiny Hut has a spherical AoE, so:
You select a sphere's point of origin, and the sphere extends outward from that point.
A spell's effect expands in straight lines from the point of origin. If no unblocked straight line extends from the point of origin to a location within the area of effect, that location isn't included in the spell's area. To block one of these imaginary lines, an obstruction must provide total cover.
To be clear, there is no RAW definition of above, so we interpret it in the usual way that above is used in English.
Relevant definitions of above all read similarly to this one:
above: At a higher level or layer than.
To an observer watching tiny hut be cast, this sense of "at a higher level or layer than" is going to be completely irrelevant to the caster's orientation.
More importantly, the caster of tiny hut should agree with this observer. If you are standing, there is a natural notion of above that you and someone observing you agree upon. Then, suppose you stand on your head. Again, you and the observer are going to agree that your feet are now above your head, and neither of you will believe that the ground is now above you and below the observer. But why is this the case?
The spell description is meta-knowledge, and trying to leverage it to get tiny hut cast upside down is metagaming. The caster does not know that the spell description text says above - this is strictly player knowledge.
When the blue character casts tiny hut, the result is:
Meta-knowledge of the spell description and pursuant rule-lawyering is the only reason to believe that standing on your head and casting tiny hut would give any result besides:
Our blue caster has no reason to disagree with his red friend when his red friend says: "above means higher than my head, even when you are standing on yours".
The only reason blue could have to disagree is that they read page 225 of the Player's Handbook and thinks that the magical weave cares that he is arbitrarily defining above to mean whatever direction his head is oriented.