Say we have some probability density function $f(x)$ which is unimodal (with the modus at 0) and also symmetric in the y-axis. It seems to me that 'in most cases', the square of a r.v $x \sim f(x)$ would follow a distribution with a positive skew. If we take the normal distribution for example, we see that taking the square of $x \sim N(0,1)$ most frequently leads to values close to 0, but can also (less frequently) lead to large positive values, indicating a positively skewed distribution.
My question is the following: does the square of a r.v $x \sim f(x)$ always follow a distribution with a positive skew, and if so, how can this be shown? Alternatively, if this does not always hold, are there conditions that ensure this property does hold?