  We study twisted products $H=\alpha^rH_r$ of natural autonomous Hamiltonians
$H_r$, each one depending on a separate set, called here separate $r$-block, of
variables. We show that, when the twist functions $\alpha^r$ are a row of the
inverse of a block-St\"ackel matrix, the dynamics of $H$ reduces to the
dynamics of the $H_r$, modified by a scalar potential depending only on
variables of the corresponding $r$-block. It is a kind of partial separation of
variables. We characterize this block-separation in an invariant way by writing
in block-form classical results of St\"ackel separation of variables. We
classify the block-separable coordinates of $\mathbb E^3$.
