Exploratory methods for scaling text, votes, and other count data are widely used for modeling policy preferences and their common logic is naturally extended to more than one policy dimension. It is well known amongst practitioners that the substantive interpretation of induced dimensions requires care and substantive knowledge. Less well known is that the orthogonality constraints necessary to model multiple dimensions force a predictable but entirely artifactual geometrical structure into position estimates that may easily be mistaken for substantive variation. This paper characterizes the nature of this geometrical artifact, suggests when it may be expected to appear, and offers a range of statistical and substantive remedies, with particular attention to confirmatory versions of the same scaling models that ground position estimates in observed document or speaker covariates.