  Let $f\colon\mathbb{T}^d\to\mathbb{T}^d$ be an Anosov diffeomorphism whose
linearization $A\in{\rm GL}(d,\mathbb{Z})$ is irreducible. Assume that $f$ is
also absolutely partially hyperbolic where a weak stable subbundle is
considered as the center subbundle. We show that if the strong stable and
unstable subbundles are jointly integrable, then $f$ is dynamically coherent
and all foliations match corresponding linear foliation under the conjugacy to
the linearization $A$. Moreover, $f$ admits the finest dominated splitting in
weak stable subbundle with dimensions matching those for $A$, and it has
spectral rigidity along all these subbundles.
  In dimension 4 we are also able to obtain a similar result which allows to
group the weak stable and unstable subbundles into a center subbundle and
assumes joint integrability of strong stable and unstable subbundles. As an
application, we show that for every symplectic diffeomorphism $f\in{\rm
Diff}^2_{\omega}(\mathbb{T}^4)$ which is $C^1$-close to an irreducible
non-conformal automorphism $A\in{\rm Sp}(4,\mathbb{Z})$, the extremal
subbundles of $f$ are jointly integrable if and only if $f$ is smoothly
conjugate to $A$.
