  We define notions of direction $L$ ergodicity, weak mixing, and mixing for a
measure preserving $\mathbb Z^d$ action $T$ on a Lebesgue probability space
$(X,\mu)$, where $L\subseteq\mathbb R^d$ is a linear subspace. For $\mathbb
R^d$ actions these notions clearly correspond to the same properties for the
restriction of $T$ to $L$. For $\mathbb Z^d$ actions $T$ we define them by
using the restriction of the unit suspension $\widetilde T$ to the direction
$L$ and to the subspace of $L^2(\widetilde X,\widetilde \mu)$ perpendicular to
the suspension rotation factor. We show that for $\mathbb Z^d$ actions these
properties are spectral invariants, as they clearly are for $\mathbb R^d$
actions. We show that for weak mixing actions $T$ in both cases, directional
ergodicity implies directional weak mixing. For ergodic $\mathbb Z^d$ actions
$T$ we explore the relationship between directional properties defined via unit
suspensions and embeddings of $T$ in $\mathbb R^d$ actions. Genericity
questions and the structure of non-ergodic and non-weakly mixing directions are
also addressed.
