Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $f$--vectors and checking the validity of the following five conjectures: Bárány, unimodality, $3^d$, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension $d$, an isocanted alcoved polytope has $2^{d+1}-2$ vertices, its face lattice is the lattice of proper subsets of $[d+1]$ and its diameter is $d+1$. They are realizations of $d$--elementary cubical polytopes. The $f$--vector of a $d$--dimensional isocanted alcoved polytope attains its maximum at the integer $\lfloor d/3\rfloor$.