Dan gave several definitions of "orientation", all confined to the case k = n in the way that everybody else seems to have interpreted the word, and all mentioning 2 connected components. It's now evident that a major source of confusion was that [coming from a computational geometry direction --- orientation?] I'm concerned rather with orientation of a subspace within the whole space. If we consider more general geometries --- Moebius / conformal / Poincar\'e, Lie sphere / physicist's conformal, and so on --- the Lie group corresponding to Euclidean isometries has 4 connected components rather than 2. On the other hand, orientation of subspaces does not appear to be any more difficult than before. So the idea occurs to me that perhaps the solution to my problem is to consider instead a 4-valued "sub-orientation"? Why should this be justified? WFL