Symmetric probability distribution

A real probability distribution is symmetric about $m$ when $X-m$ and $m-X$ have the same distribution. A centered normal distribution is symmetric about zero.

A probability distribution on a vector space is centrally symmetric about zero when $X$ and $-X$ have the same distribution. Independent observations from such a distribution are jointly invariant under coordinatewise sign changes.

A Rademacher random variable takes the values $1$ and $-1$, each with probability $1/2$.