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.
The Gambler’s Fallacy applies in both directions: if an event has happened more frequently in the past than expected, then the Fallacy states that it is less likely to occur in future as well. So for example, rolling a 6-sided die three times and getting two sixes in such a world should also decrease the probability of getting another six on the next roll by some unspecified amount.
That is, it’s a world in which steps in every random walk are biased toward the mean.
However, that does run into some difficulties. Suppose that person A is flipping coins and keeping track of the numbers of heads and tails. The count is 89 tails to 111 heads so far. Person B comes in watches for 100 more flips. They see 56 more tails and 44 heads, so that A’s count is now at 145 tails to 155 heads. Gambler’s Verity applied to A means that tails should still be more likely. Gambler’s Verity applied to B means that heads should be more likely. Which effect is stronger?
Now consider person C who isn’t told the outcomes of each flip, just whether the flip moved the counts more toward equal or further away. Gambler’s Verity for those who see each flip means that “toward equal” flips are more common than “more unequal” flips. But applied to C’s observations, Gambler’s Verity acts to cancel out any bias even more rapidly than independent chance would. So if you’re aware of Gambler’s Verity and try to study it, then it cancels itself out!