Orthogonal range reporting with fixed upper rectangular corner
The count of any rectangle $C(U, L, D, R)$ can be obtained combining the counts of 4 fixed "upper-left" retangles:

$C(U, L, D, R) = C(0, 0, D, R) - C(0, 0, U, R) - C(U, 0, D, L) + C(0, 0, U, L)$,

noting $C(up, left, down, right)$ the count in the rectangle delimited by $(up, left, down, right)$.

Thus, you cannot expect more than a factor 4 runtime improvement.