  An endomorphism $f$ of a projective variety X is polarized (resp.
quasi-polarized) if $f^*H$ is linearly equivalent to $qH$ for some ample (resp.
nef and big) Cartier divisor $H$ and integer $q > 1$. First, we use cone
analysis to show that a quasi-polarized endomorphism is always polarized, and
the polarized property descends via any equivariant dominant rational map.
Next, we show that a suitable maximal rationally connected fibration (MRC) can
be made $f$-equivariant using a construction of N. Nakayama, that $f$ descends
to a polarized endomorphism of the base Y of this MRC and that this Y is a
Q-abelian variety (quasi-\'etale quotient of an abelian variety). Finally, we
show that we can run the minimal model program (MMP) $f$-equivariantly for
mildly singular X and reach either a Q-abelian variety or a Fano variety of
Picard number one.
  As a consequence, the building blocks of polarized endomorphisms are those of
Q-abelian varieties and those of Fano varieties of Picard number one.
  Along the way, we show that $f$ always descends to a polarized endomorphism
of the Albanese variety Alb(X) of X, and that the pullback of a power of $f$
acts as a scalar multiplication on the Neron-Severi group of X (modulo torsion)
when X is smooth and rationally connected.
  Partial answers about X being of Calabi-Yau type, or Fano type are also given
with an extra primitivity assumption on $f$ which seems necessary by an
example.
