There are two natural ways to symmetrize a real-valued random variable X:

• Multiplying X by an independent random sign S to obtain SX.
• Taking X-X', where X' is an independent replica of X.

In general, these have different (symmetric) distributions.
Sometimes they are equal; for example:

1. X is {0,1}-valued with equal probabilities.
2. X is exponentially distributed (i.e. P(X>x)=e^-x, x>0).
3. ???...

Off hand, it is not easy to produce other examples (except for scale change of the above).

In the one-sided case (nonnegative X), equality of the natural symmetrizations means that X and |X-X'| are equally distributed.
A full characterization of all distributions for X in the one-sided case will be presented.
In particular, it will be shown that the exponential distribution is the only non-lattice distribution for which equality is obtained (in the one-sided case). All the lattice distributions which satisfy the equality (again --- in the one-sided case only) will also be exhibited.

The general (two-sided) case of the problem is still widely open. There are some ad-hoc sporadic examples, but a general pattern has yet to emerge.