Can we transmit FM signal using AM band and AM signal using FM band?
AM now supports Quadrature Stereo AM but still fits within 10Khz channel spacing with some guardband.

FM cannot fit into the AM band , wwhile AM is even used in the ISM band at 928MHz with 6kHz channel spacing, so there is no reason why AM cannot be used at any frequency.

FM has higher SNR and signal bandwidth that prevents application in the lower AM bands. 

However one can use AM or FM for cable communication at any frequency.

Can you see how to fit the above baseband FM signal in the 10KHz channel spacing of the AM band?  If you do it's worth a ton of money.