That's an odd situation as described and seemingly caused by two faults:
1) The r/hand dash repeater bulb may be blown since it never works on hazards or when right indicator selected. Check wiring to it as well though.
2) The selector switch when right selected is not functioning. Look for green/white lead from same checking switch and connection to same.
Note although the hazard switch with it's own flasher unit are common sources of indicator issues, as described this part of circuitry seems to be working correctly and allowing left indicator to be selected with hazards off.
