  In this paper, we will consider regularity criteria for the Navier--Stokes
equation in mixed Lebesgue sum spaces. In particular, we will prove regularity
criteria that only require control of the velocity, vorticity, or the positive
part of the second eigenvalue of the strain matrix, in the sum space of two
scale critical spaces. This represents a significant step forward, because each
sum space regularity criterion covers a whole family of scale critical
regularity criteria in a single estimate. In order to show this, we will also
prove a new inclusion and inequality for sum spaces in families of mixed
Lebesgue spaces with a scale invariance that is also of independent interest.
