HMMT November 2018 T7

A $5 \times 5$ grid of squares is filled with integers. Call a rectangle corner-odd if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid? Note: A rectangle must have four distinct corners to be considered corner-odd; i.e. no $1 \times k$ rectangle can be corner-odd for any positive integer $k$.