A general, not necessarily stochastic, kernel P(x, A) is said to be
reversible
with respect to a measure 
if for every 
the integral
is symmetric under the interchange of u and v. Equivalently,
the operator T on 
defined by
is self-adjoint. This is the definition of `reversible' preferred by
those who like to think of probabilities as positive linear operators
on function spaces rather than as set functions on 
-fields.
Those who like probabilities
as set functions use the same definition with u and v replaced by
indicators of sets, that
is symmetric under the interchange of A and B. The two views are,
of course, equivalent by extension by linearity and monotone convergence.
When P is stochastic, the special case 
gives
so 
and is stationary for P. Thus verification of
reversibility of a stochastic kernel P with respect to is a tool
for establishing stationarity of for P.