There is a general notion of Symmetry in mathematics, that of an object being constant under some transformation.
If we think of our object as being a "mathematical problem" we can see certain mathemtical problems have symmetries in them, in terms of how the problem is stated.
This thought came to be when I was trying to solve a combinatorics problem on this website.
The problem statement was "Given 11 people and one safe, how many locks do we need on the safe to ensure a subset of 5 people cannot open the safe but a susbet of 6 people can open the safe. For each lock we can get as many copies of its key as we like.".
Now This is just an example, I am not asking about this problem in particular.
I merely want to note that in this problem the group of 11 people all have the same constraints put on them.
There is no mentioning of a property fulfilled by a certain "distinguished" subset of these 11 people, if we were to write the problem as:
"Given a set of 11 people $\{p_1,\dots ,p_{11}\}$, how many locks do we need ..." (the rest of the problem statement is the same as before),
then we would have a symmetry in the sense that if we permute the set $\{p_1,\dots ,p_{11}\}$ in the problem statement, and relabel the people accordingly, the problem statement does not change.
Intuitively, I expect a solution to such a problem to be symmetric in some way, for example I would expect each person do have the same number of keys. Intuition is not a proof.
The question is this: Is there a general principle in mathematics which relates symmetries in problem statemenets to symmetries in the solution?
It is not in general true that a solution to a symmetric problem must itself be symmetric. For example:
This is an even function, so it is symmetric in reflection about the $y$ axis. However, the solutions are $1$ and $-1$, and neither of these solutions is itself symmetric. (Physicists speak about "symmetry breaking" in such cases, and it more complex variants it plays a significant role in modern particle physics).
What we can say is that if there's a symmetry of the problem, we can apply that same symmetry to the solution, and get a (possibly different) solution back. That's more or less the generic definition of what it means for a problem to have a symmetry. So the set of all solutions will be a symmetric as the problem is.
If we know somehow that the solution is unique, it will need to have all of the symmetries the problem has. This is sometimes a useful shortcut for finding it. (But be careful to check whether "unique" really means "unique modulo such-and-such symmetries which are present in the problem too").
In the vein of your example we could say
That's kind of a silly problem, but it's certainly symmetric under permutation of the two people. But the solution does not have the kind of symmetry you're intuitively expecting: One lock with one key, and hand the single key to one of the trusted people.