Is there an estimator for the symmetry of a bimodal distribution? I would like to know how I can measure the degree of symmetry of a bimodal distribution.
Is there any a criterion like, for example skewness, in the case of unimodal distributions?
 A: By definition, a symmetric random variable $X$ is one for which there is a constant $\mu$ for which $X-\mu$ and $\mu-X$ are identically distributed.  In terms of the distribution function $F$ this is equivalent to
$$\eqalign{
F(\mu+x) &= \Pr(X \le \mu+x) = \Pr(X-\mu \le x) \\
 &= \Pr(\mu-X \le x) \\
 &= \Pr(X-\mu \ge -x) = \Pr(X \ge \mu - x)\\
 &= 1 - \Pr(X \lt \mu- x) \\
 &= 1 - F(\mu - x) + \Pr(X = \mu-x)
}$$
When $F$ is continuous this simplifies to 
$$F(\mu + x) + F(\mu-x) = 1$$
for all $x$ and thence (via differentiation), when $F$ is absolutely continuous with distribution function $f$,
$$f(\mu + x) = f(\mu - x)$$
for all $x$.  (It's not hard to see that any of these equations uniquely determine $\mu$.)
This provides a general, flexible procedure to test for symmetry, based on any method of comparing two distributions.  (Many such methods exist, ranging from comparing basic properties like moments through relative entropy, KL distances, and so on.)  Specifically, take any non-negative function $\delta$ where
$$\delta(F,G)$$
is intended to measure the "distance" or "dissimilarity" between distributions $F$ and $G$.  All we ask of $\delta$ (besides being nonnegative) is that $\delta(F,G) = 0$ if and only if $F=G$.
For any constant $\mu$ define $F_\mu(x) = F(x-\mu)$  and $\check F_\mu(x) = F(\mu-x)$.  Then merely take
$$\inf_{\mu}\, (\delta(F_\mu, \check{F}_\mu))$$
as the measure of asymmetry. This measures how close you can make $X-\mu$ and $\mu-X$ appear to be.   It will be nonnegative and equal to zero only when $F$ is symmetric.  Choose $\delta$ to emphasize those aspects of "closeness" important in your application, such as asymptotic tail behavior or balancing an odd moment.
As a simple example, intended to be applied to bimodal absolutely continuous distributions, let
$$\delta(F,G) = \int (f(x) - g(x))^2 dx.$$
This is the $L^2$ norm of their density functions, depending on the total area between their graphs.  The illustration shows the density $f$ for a bimodal distribution at the left, followed by three graphs depicting the region between $f_\mu$ and $\check{f}_\mu$ for values of $\mu$ around the optimum $\mu=1$, where that region is the smallest in the $L^2$ sense:

This shows how the whole process lends itself to exploratory (visual) evaluation: simply make such a plot to superimpose $f_\mu$ and $\check{f}_\mu$ (or the CDFs $F_\mu$ and $\check{F}_\mu$) and vary $\mu$ until the graphs look as "alike" as possible.  The visual deviations at this optimal point will not only indicate asymmetry, but they will also show the form of the asymmetry.
A: While I don't think that there is a single measure of symmetry for a bimodal distribution in general, for a special case of mixture of two normal distributions, perhaps, it is possible to use one or several bi-modality measures and statistical tests. Some mixture modeling software (especially, some R packages) might have some of those measures assessment and tests implemented, so that it might be possible to determine the level of symmetry of a bimodal distribution analytically.
