Also see this:
There are actually 6 combinations so only 2^42.666666...7 unique combos.
You have just discovered an endomorphism, where two different Y's will solve the curve equation: y^2 = x^3 + 7
i.e. these two privkeys make two different but opposite Y's (because 115792089237316195423570985008687907852837564279074904382605163141518161494336 is just n-1 or, equivalently [mod n: the cyclic group 0..n, n-1...2n, etc.], it is -1).
And -1^2 = 1^2 = 1.
Now the other 3 combinations - and why only 3? see the next section - come from the X term.
Notice how the X is cubed which means it has three different roots if you consider it as a polynomial. There's obviously X, but there's also 0+Xi and 0-Xi (complex numbers). It follows the pattern [X + Yi], where the Y coord is an imaginary number.
This goes to say that if e.g. (7,0) was a valid point, then that, (0,7) and (0,n-7) would all reference similar points.
And (x,y), (y,x) and (y, n-x) would similarly reference similar points as well.
Now multiply 2*3 combos (endomorphisms) and you get a total of 6 endomorphisms: (x,y), (y,x) (y, n-x) and (x,n-y), (n-y, x), (n-y, n-x).
It isn't something like they'd all have the same Y-point, but these points are accessible from the same X-coordinate as well. (See e.g.
Roots of x^3+7 example)
There is 6-way symmetry on secp256k1 not just 2-way. Y-coordinate symmetry accounts for 3x the mirror symmetry you're talking about (which is 2).
