  In a cylinder $D_T = \Omega \times (0,T)$, where $\Omega\subset
\mathbb{R}^n$, we examine the relation between the $L$-caloric measure,
$d\omega^{(x,t)}$, where $L$ is the heat operator associated with a system of
vector fields of H\"ormander type, and the measure $d\sigma_X\times dt$, where
$d\sigma_X$ is the intrinsic $X$-perimeter measure. The latter constitutes the
appropriate replacement for the standard surface measure on the boundary and
plays a central role in sub-Riemannian geometric measure theory. Under suitable
assumptions on the domain $\Omega$ we establish the mutual absolute continuity
of $d\omega^{(x,t)}$ and $d\sigma_X\times dt$. We also derive the solvability
of the initial-Dirichlet problem for $L$ with boundary data in appropriate $
L^p$ spaces, for every $p>1$.
