Orientations of objects in $\mathbb{R}^p$ with symmetry group $K$ cannot be described unambiguously by elements of the rotation group $SO(p)$, but correspond instead to elements of the quotient space $SO(p)/K$. Specifications of probability distributions and appropriate statistical methods for such objects have been lacking -- with the notable exceptions of axial objects in the plane and in $\mathbb{R}^3$.

We exploit suitable embeddings of $SO(p)/K$ into spaces of symmetric tensors to provide a systematic and intuitively appealing approach to the statistical analysis of the orientations of such objects. We firstly consider the case of {\em orthogonal $r$-frames} in $\mathbb{R}^p$, corresponding to sets of $r\leq p$ mutually orthogonal axes in $p$ dimensions, which includes the Watson and Bingham distributions. Using the same approach we then treat the three dimensional case of $SO(3)/K$, where $K$ is one of the point symmetry groups. In both cases the resulting tools include measures of location and dispersion, tests of uniformity, one-sample tests for a preferred orientation and two-sample tests for a difference in orientation. The methods are illustrated using data from earthquake focal mechanisms (fault plane and slip vector orientations) and crystallographic data with orientation measurements of distinct mineral phases.