It is more complicated than that, unfortunately. After reading some papers, I concluded that the monopole should behave just like a dipole, since the ground plane acts as a "mirror" creating a mirror image of the monopole. And that a dipole has an electrical center right near where the two elements join the middle, since the dipole antenna must look like a dipole of two charges (+ and -).
Then I actually performed this simple measurement below. I used a halfwave dipole at the receiving end, and a monopole at the transmitting end. I used a network analyzer and an RF amp to boost the signal level. Antennas were 3 meters apart, and tuned to 885 MHz. With the monopole vertical, I recorded path loss and phase and called them the references (i.e. 0 dB and 0 degrees). Then I rotated the monopole so that it was pointing toward the dipole, as shown below. I expected to see the signal null out (assuming it would be cross polarized), and assumed it would stay at roughly 0 degrees of phase shift. The data shows it does NOT null out, and the phase shift is huge! Not at all what I expected! So I guess I do not understand this phenomenon at all.
What really shocks me is the phase shift. If the monopole phase center was somewhere along the monopole whip, the longest it could be away from the rotating point would be 90 degrees. Yet During rotation I got 169 degrees! Seems physically impossible! The rotating point was chosen so the point where the whip joins the ground plane would remain 3 meters from the receiving antenna.
Oddly enough, there was a gain peak (+3.4 dB) with the elevation of the monopole at 45 degrees.
Some factors I have that might be influencing the data:
It is a small ground plane, ie around 1/2 wavelength in diameter
There might be some multipath, as it was not done in a proper anechoic chamber, and it is too snowy to do it outside this month.
Added after 24 minutes:
Here are what the antennas looked like: