If you're in a classical-physics simulation that's guaranteed to have rotational symmetry, then yep, you stay mirrored.

In any messy real world setting (even ignoring quantum mechanics), chaos theory would indeed kick in pretty quickly - the thermal jostling of air molecules, or radiation from outside, would slowly lead to neurons firing at slightly different times.

But maybe the most sciencey way to unmirror yourselves is to suppose you have the ability to prepare and measure an entangled quantum state. E.g. suppose you meet in the middle and put the spin of a pair of atoms into the state |up,down>+|down,up>. This state is rotationally symmetric (for the nitpickers, I think I've implied the atoms have integer spin), so you can do this, but when you measure the atoms you'll get opposite results.

In our universe’s physics, the symmetry breaks immediately and your clone possibly dies, see https://en.wikipedia.org/wiki/Chirality

If this room is still on Earth (or on any other rotating body), you could in principle set up a Foucault pendulum to determine which way the rotation is going, which breaks mirror symmetry.

If the room is still in our Universe, you can (with enough equipment) measure any neutrinos that are passing through for helicity "handedness". All observations of fusion neutrinos in our universe are left-handed, and these by far dominate due to production in stars. Mirror transformations reverse helicity, so you will disagree about the expected result.

If the room is somehow isolated from the rest of the universe by sufficiently magical technology, in principle you could even wait for long enough that enough of the radioactive atoms in your bodies and the room decay to produce detectable neutrinos or antineutrinos. By mirror symmetry the atoms that decay on each side of the room are the same, and so emit the same type (neutrino or antineutrino, with corresponding handedness). You would be waiting a long time with any known detection methods though.

This would fail if your clone's half room was made of antimatter, but an experiment in which half the room is matter and half is antimatter won't last long enough to be of concern about symmetry. The question of whether the explosion is mirror-symmetric or not will be irrelevant to the participants.

Assuming you're somehow gravitationally/rotationally symmetrical as well, and that quantum uncertainty doesn't matter, you are probably right.  I strongly suspect that this pile of assumptions is not just infeasible, but impossible in our current universe.

I think the classic answer to the "Ozma Problem" (how to communicate to far-away aliens what earthlings mean by right and left) is the Wu experiment.  Electromagnetism and the strong nuclear force aren't handed, but the weak nuclear force is handed.  Left-handed electrons participate in weak nuclear force interactions but right-handed electrons are invisible to weak interactions^[1].

(amateur, others can correct me)

1. ^{^}

   Like electrons, right-handed neutrinos are also invisible to weak interactions.  Unlike electrons, neutrinos are also invisible to the other forces*^[2].  So the standard model basically predicts there should invisible particles wizzing around everywhere that we have no way to detect or confirm exist at all.  

2. ^{^}

   Besides gravity

The premise seems like a hard ask, though if we assume precise control over all shapes on all 'observable' levels, I feel there is always background noise, radiation of different kinds, so when you are not in the same position there will be slightly different values reaching you.[edit: so when this reaches biological systems it will always affect them slightly. So you could probably just sit there, only if we are looking for an observable threshold you might be there a while depending on levels]

1. As a human, you react to what other humans do around you. Your reaction algorithms may not allow two copies of you to mirror each other.
2. The clone is not mirrored, so you are not in front of a mirror you, but a rotated you.

Maybe you could treat each other as two parts of the same system. It hurts to pull your hair out so suppose you had some scissors too. You could cut your hair, and scatter them on the floor. Wind is really sensitive to initial conditions (consider brownian motion) so suppose you blow the hair and you both decide to pick them up. Since you're different distances to the hair strands you could break the symmetry?

Step outside of the room. You'll see the world from different locations, symmetry will be broken. IDK why you're interested in a situation that can't happen, perfect sensory symmetry for long periods.

Free will is a contradiction in terms, so I don't think it's worth fussing over exactly that wording. What people ususally mean by free will is that their decisions truly affect the future. If you take "they" to refer to their brain state, this is pretty clearly true on that interpretation. I think other definitions of self are incoherent and that one is pretty straightforwardly what we mean by "me", so I agree that "free will" in its common definition is compatible with determinism.

Just physically interact. Push each other around. You’ll be building up tiny differences, and those interactions will magnify those differences. 

Also, smash the environment. I originally read you post as the room had mirrors for walls, and that’s what made me think of it. 

I don’t think that your question is quite makes sense. The world is non-deterministic. There are macroscopic patterns that are generally symmetrical, but not at the deepest levels. For instance, there is the cosmic gravitational background, where space is sort of wobbling around because of the gravitational waves from other things in the universe moving about, similarly to ripples on a pond. Even if you controlled for everything in the room, you could not control for those differences. The only way for that room to be perfectly rotationally symmetrical, is if the universe is rotationally symmetrical relative to that room.

THEN you can talk about quantum field theory. 

Given your clone is a perfectly mirrored copy of yourself down to the lowest physical level (whatever that means), then breaking symmetry would violate the homogeneity or isotropy of physics. I don't know where the physics literature stands on the likelihood of that happening (even though certainly we don't see macroscopic violations).

Of course, it might be an atom-by-atom copy is not a copy down to the lowest physical level, in which case trivially you can get eventual asymmetry. I mean, it doesn't even make complete sense to say "atom-by-atom copy" in the language of quantum mechanics, since you can't be arbitrarily certain about the position and velocity of each atom. Maybe saying something like "the quantum state function of the whole room is perfectly symmetric in this specific way". I think then (if that is indeed the lowest physical level) the function will remain symmetric forever, but maybe in some universes you and your copy end up in different places? That is, the symmetry would happen at another level in this example: across universes, and not necessarily inside each single universe?

It might also be there is no lowest physical level, just unending complexity all the way down (this had a philosophical name which I now forget).