The symmetric square of a topological space $X$ is obtained from the usual square $X^2$ by identifying pairs of symmetric points $(x_1,x_2)$ and $(x_2,x_1)$. Thus, elements of the symmetric square can be identified with unordered pairs $\{x_1,x_2\}$ from $X$ (including the degenerate case $x_1 = x_2$). With this identification, the following question is very natural:

When is there a continuous selector from the symmetric square to the original space?

This is always possible if $X$ is a generalized ordered space (a subspace of a linear order with the order topology) since min and max are both continuous selectors. I don't think this is a necessary condition since I can construct continuous selectors (on Baire space, for example) that aren't equal to min or max for any ordering of the underlying space.

A necessary condition is that removing the diagonal disconnects $X^2$. More precisely, as Adam Bjorndahl pointed out in the comments, the complement of the diagonal should be the union of two disjoint open sets such that $(x_1,x_2)$ is in one iff $(x_2,x_1)$ is in the other. So, for example, there are no continuous selectors for the symmetric square of the plane nor the circle.

I'm mostly interested in the case when $X$ is Polish, so it's fine to assume that $X$ is very nice: Hausdorff, normal, perfect, etc.