For some reason, they don't teach you this stuff in school, or else, I
wasn't paying attention that day :-)
I found some even-order non tensor-product rules for quadrilateral
elements. These rules contain fewer evaluation points than the
equivalent-order tensor product rules. Here is the reference:
J. W. Wissmann and T. Becker, "Partially symmetric cubature
formulas for even degrees of exactness," SIAM J. Numer. Anal. 23
I implemented and tested the degree 4, 6, and 8 rules from the paper.
They have 6, 10, and 16 points, respectively, while the equivalent
tensor product rules had 9, 16, and 25 points, respectively. I
believe this has something to do with the odd-order of the 1D Gauss
rules (N points integrate polynomials up to degree 2N-1 exactly) so
that 4th and 5th order get lumped together, 6th and 7th, etc.
Wissmann's degree 4 and 6 rules are even provably minimal for this
element type and order. I also found the arrangement of the points,
when plotted, to be quite interesting. A plot of the points in the
pair of degree-6 rules (x's for rule 1, o's for rule 2) obtained by
Wissmann is attached.