  A \emph{mixed interval graph} is an interval graph that has, for every pair
of intersecting intervals, either an arc (directed arbitrarily) or an
(undirected) edge. We are particularly interested in scenarios where edges and
arcs are defined by the geometry of intervals. In a proper coloring of a mixed
interval graph $G$, an interval $u$ receives a lower (different) color than an
interval $v$ if $G$ contains arc $(u,v)$ (edge $\{u,v\}$). Coloring of mixed
graphs has applications, for example, in scheduling with precedence
constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general
mixed interval graphs, we present a $\min \{\omega(G), \lambda(G)+1
\}$-approximation algorithm, where $\omega(G)$ is the size of a largest clique
and $\lambda(G)$ is the length of a longest directed path in $G$. For the
subclass of \emph{bidirectional interval graphs} (introduced recently for an
application in graph drawing), we show that optimal coloring is NP-hard. This
was known for general mixed interval graphs. We introduce a new natural class
of mixed interval graphs, which we call \emph{containment interval graphs}. In
such a graph, there is an arc $(u,v)$ if interval $u$ contains interval $v$,
and there is an edge $\{u,v\}$ if $u$ and $v$ overlap. We show that these
graphs can be recognized in polynomial time, that coloring them with the
minimum number of colors is NP-hard, and that there is a 2-approximation
algorithm for coloring.
