  The spin of a single electron confined in a semiconductor quantum dot is a
natural qubit candidate. Fundamental building blocks of spin-based quantum
computing have been demonstrated in double quantum dots with significant
spin-orbit coupling. Here, we show that spin-orbit-coupled double quantum dots
can be categorised in six classes, according to a partitioning of the
multi-dimensional space of their $g$-tensors. The class determines physical
characteristics of the double dot, i.e., features in transport, spectroscopy
and coherence measurements, as well as qubit control, shuttling, and readout
experiments. In particular, we predict that the spin physics is highly
simplified due to pseudospin conservation, whenever the external magnetic field
is pointing to special directions (`magic directions'), where the number of
special directions is determined by the class. We also analyze the existence
and relevance of magic loops in the space of magnetic-field directions,
corresponding to equal local Zeeman splittings. These results present an
important step toward precise interpretation and efficient design of spin-based
quantum computing experiments in materials with strong spin-orbit coupling.
