  Let $I(X,K)$ be the incidence algebra of a finite connected poset $X$ over a
field $K$ and $D(X,K)$ its subalgebra consisting of diagonal elements. We
describe the bijective linear maps $\varphi:I(X,K)\to I(X,K)$ that strongly
preserve the commutativity and satisfy $\varphi(D(X,K))=D(X,K)$. We prove that
such a map $\varphi$ is a composition of a commutativity preserver of shift
type and a commutativity preserver associated to a quadruple
$(\theta,\sigma,c,\kappa)$ of simpler maps $\theta$, $\sigma$, $c$ and a
sequence $\kappa$ of elements of $K$.
