When you describe this kind of key relationship, I just "get it," because I know the key signatures well enough. But I really wanted to know why or how this relationship exists.
Below is a chart I drew showing the circle of fifths (and circle of fourths) but arranged as a line. I'm not sure this will help you, but it shows the key 'symmetries' with keys paired up according to the funny modulus math @Richard describes in his answer.
I like visuals and made the chart for myself last night. (I couldn't get to sleep after reading this post and had to make the chart just to clear my head!) Maybe it will be helpful or interesting to you...
The middle number row 0-11 is both the position on the circle of fifths/fourths and the number of sharps or flats in the key signature.
When you go past the two dashed line boundaries - beyond the area I labelled the _7 letter gamut - you get into the weird enharmonically complex keys. You can get the modulus math to work in this region by adding a _negative number (bottom row) which represents going backwards on the circle of fifths. Ex.
E#/F at position 11 and
Fb/E at position -4, 11 + -4 = 7.
When all 12 chromatic pitch classes are arranged by perfect fifths and numbered 0-11, the basic sequence for a major scale - starting on C - is
0,2,4,11,1,3,5. All the other keys signatures are transpositions of that numeric sequence on the circle. For example, just add 2 to each position to get
D major (remember that 11 will circle back around to 0 the 1.)
At the bottom of the chart is a sort of test that the numbering in the enharmonically complex area outside the gamut is true. As bizarre as it is to say 'Eb major is 9 sharps' is mathematically true.
Finally, the is a kind of geometric symmetry of scale for each position along the circle. Any major key in the sharp keys shares mirror symmetry with the corresponding flat key's fifth mode of the relative minor. It's easier to explain by example: at position 3
A ascending shares a mirror symmetry with
C minor (
Eb's relative minor)
G descending. You can test that on the keyboard as see how the two scales are inversions sharing the same sequence of whole and half steps and the same sequence of white and black keys.
There isn't a specific musical term for all this, but when you write out the circle of fifths as a circle of counting line or compare the inversions of scales you have a kind of mathematical symmetry.