A random variable [math] is called an independent symmetrizer of a given random variable [math] if (a) it is independent of [math] and (b) the distribution of [math] is symmetric about [math] . In cases where the distribution of [math] is symmetric about its mean, it is easy to see that the constant random variable [math] is a minimum-variance independent symmetrizer. Taking [math] to have the same distribution as [math] clearly produces a symmetric sum, but it may not be of minimum variance. We say that a random variable [math] is symmetry resistant if the variance of any symmetrizer, [math] , is never smaller than the variance of [math] . Let [math] be a binary random variable: [math] and [math] where [math] , [math] , and [math] . We prove that such a binary random variable is symmetry resistant if (and only if) [math] . Note that the minimum variance as a function of [math] is discontinuous at [math] . Dropping the independence assumption, we show that the minimum variance reduces to [math] , which is a continuous function of [math] .