  Let $\phi: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected
manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$,
denote by $\xi\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let
$\rho:\pi_1(X)\to GL(n,\mathbb{C})$ be a representation, denote by
$H^*(X,\rho)$ the corresponding twisted cohomology of $X$. Denote by $\rho_0$
the restriction of $\rho$ to $\pi_1(M)$, and by $\rho^*_0$ the
antirepresentation conjugate to $\rho_0$. We construct from these data an
automorphism of the group $H_*(M,\rho^*_0)$, that we call the twisted monodromy
homomorphism $\phi_*$. The aim of the present work is to establish a relation
between Massey products in $H^*(X,\rho)$ and Jordan blocks of $\phi_*$. Given a
non-zero complex number $\lambda$ define a representation
$\rho_\lambda:\pi_1(X)\to GL(n,\mathbb{C})$ as follows:
$\rho_\lambda(g)=\lambda^{\xi(g)}\cdot\rho(g)$. Denote by $J_k(\phi_*,
\lambda)$ the maximal size of a Jordan block of eigenvalue $\lambda$ of the
automorphism $\phi_*$ in the homology of degree $k$.
  The main result of the paper says that $J_k(\phi_*, \lambda)$ is equal to the
maximal length of a non-zero Massey product of the form $\langle \xi, \ldots ,
\xi, x\rangle$ where $x\in H^k(X,\rho)$ (here the length means the number of
entries of $\xi$). In particular, $\phi_*$ is diagonalizable, if a suitable
formality condition holds for the manifold $X$. This is the case if $X$ a
compact K\"ahler manifold and $\rho$ is a semisimple representation. The proof
of the main theorem is based on the fact that the above Massey products can be
identified with differentials in a Massey spectral sequence, which in turn can
be explicitly computed in terms of the Jordan normal form of $\phi_*$.
