  For any continuous zero-mean random variable (r.v.) X, a reciprocating
function r is constructed, based only on the distribution of X, such that the
conditional distribution of X given the (at-most-)two-point set {X,r(X)} is the
zero-mean distribution on this set; in fact, a more general construction
without the continuity assumption is given in this paper, as well as a large
variety of other related results, including characterizations of the
reciprocating function and modeling distribution asymmetry patterns. The
mentioned disintegration of zero-mean r.v.'s implies, in particular, that an
arbitrary zero-mean distribution is represented as the mixture of two-point
zero-mean distributions; moreover, this mixture representation is most
symmetric in a variety of senses. Somewhat similar representations -- of any
probability distribution as the mixture of two-point distributions with the
same skewness coefficient (but possibly with different means) -- go back to
Kolmogorov; very recently, Aizenman et al. further developed such
representations and applied them to (anti-)concentration inequalities for
functions of independent random variables and to spectral localization for
random Schroedinger operators. One kind of application given in the present
paper is to construct certain statistical tests for asymmetry patterns and for
location without symmetry conditions. Exact inequalities implying conservative
properties of such tests are presented. These developments extend results
established earlier by Efron, Eaton, and Pinelis under a symmetry condition.
