We show that an asymmetric Bernoulli random variable is symmetry resistant in the sense that any independent random variable, which when added to it produces a symmetric sum, must have a variance at least as much as itself. The main instrument is to use Skorokhod embedding to transfer the discrete problem to the realm of stochastic calculus.
"Symmetrization of Bernoulli." Electron. Commun. Probab. 13 194 - 197, 2008. https://doi.org/10.1214/ECP.v13-1364