  We consider tessellations of the Euclidean $(d-1)$-sphere by
$(d-2)$-dimensional great subspheres or, equivalently, tessellations of
Euclidean $d$-space by hyperplanes through the origin; these we call conical
tessellations. For random polyhedral cones defined as typical cones in a
conical tessellation by random hyperplanes, and for random cones which are dual
to these in distribution, we study expectations for a general class of
geometric functionals. They include combinatorial quantities, such as face
numbers, as well as, for example, conical intrinsic volumes. For isotropic
conical tessellations (those generated by random hyperplanes with spherically
symmetric distribution), we determine the complete covariance structure of the
random vector whose components are the $k$-face contents of the induced
spherical random polytopes. This result can be considered as a spherical
counterpart of a classical result due to Roger Miles.
