  We present a routing algorithm for the directed $\Theta_4$-graph, here
denoted as the $\overrightarrow{\Theta_4}}$-graph, that computes a path between
any two vertices $s$ and $t$ having length at most $17$ times the Euclidean
distance between $s$ and $t$. To compute this path, at each step, the algorithm
only uses knowledge of the location of the current vertex, its (at most four)
outgoing edges, the destination vertex, and one additional bit of information
in order to determine the next edge to follow. This provides the first known
online, local, competitive routing algorithm with constant routing ratio for
the $\Theta_4$-graph, as well as improving the best known upper bound on the
spanning ratio of these graphs from $237$ to $17$. We also show that without
this additional bit of information, the routing ratio increases to $\sqrt{290}
\approx 17.03$.
