  The order type of a point set in $R^d$ maps each $(d{+}1)$-tuple of points to
its orientation (e.g., clockwise or counterclockwise in $R^2$). Two point sets
$X$ and $Y$ have the same order type if there exists a mapping $f$ from $X$ to
$Y$ for which every $(d{+}1)$-tuple $(a_1,a_2,\ldots,a_{d+1})$ of $X$ and the
corresponding tuple $(f(a_1),f(a_2),\ldots,f(a_{d+1}))$ in $Y$ have the same
orientation. In this paper we investigate the complexity of determining whether
two point sets have the same order type. We provide an $O(n^d)$ algorithm for
this task, thereby improving upon the $O(n^{\lfloor{3d/2}\rfloor})$ algorithm
of Goodman and Pollack (1983). The algorithm uses only order type queries and
also works for abstract order types (or acyclic oriented matroids). Our
algorithm is optimal, both in the abstract setting and for realizable points
sets if the algorithm only uses order type queries.
