The best interview answers are often the shortest, and symmetry is how you get them. It is also a genuine test of understanding, because the calculation-heavy route usually gets there too - just slower and with more chances to slip.
The principle
If you can relabel the objects in a problem so that outcome A becomes outcome B and the setup is unchanged, then A and B are equally likely.
Example one
Two players alternately draw cards from a shuffled deck without replacement until someone draws the ace of spades. Is it better to go first?
The instinct is to sum a series over draw positions. Do not.
The ace of spades is equally likely to be in any of the 52 positions. Player one takes odd positions, player two takes even. There are 26 of each. So it is exactly even.
Example two
In a random permutation of 1 to n, what is the probability that 1 appears before 2?
Ignore everything else. Consider only the relative order of 1 and 2. By symmetry there is nothing distinguishing them, so 1/2.
This generalises beautifully: the probability that 1, 2 and 3 appear in that exact relative order is 1/6, because all six orderings are equally likely.
Example three
A stick is broken at two uniformly random points. What is the probability the middle piece is the longest?
By symmetry, each of the three pieces is equally likely to be the longest, so 1/3 - no integration required. Compare that with the standard "can the pieces form a triangle" question, which is genuinely 1/4 and does need work.
Where it goes wrong
Symmetry that is not actually there. In the stick problem, the three pieces are symmetric; in a variant where you break the stick, then break the longer piece, they are not, and the same reasoning gives the wrong answer.
Before invoking symmetry, say precisely what the relabelling is. If you cannot state it, you do not have it.
In the interview
Say "by symmetry" and then say which symmetry. An interviewer hearing an unjustified "1/2" cannot tell insight from a guess, and will probe until they can.