B737 Captain reported a fumes event after engine start and during gate departure. Flight returned to the gate for maintenance and the same problem occurred again during engine start.

Narrative

On taxi out a slight unusual odor was smelled in the flight deck right after engine start. The odor was similar to fuel or exhaust. Both myself and FO agreed it was from the engine start and would quickly go away. On taxi out the flight attendants (FAs) called and said there was a strong odor in the aircraft; stronger in the aft; and the FAs in the aft reported the odor was burning their nose and eyes. We immediately coordinated a gate return. Maintenance log page. No one needed any medical assistance. Maintenance troubleshooted the issue and cleared the aircraft after working on it for a few hours. Prior to boarding; Maintenance ran the aircraft packs on both the engine bleeds; both engines running; and the APU bleeds. Myself; my FO; and the FAs; walked through the cabin with Maintenance and agreed the odor was not present. We reboarded the aircraft and on engine start the second time the odor came back again. FAs called the flight deck and said passengers were complaining about the strong odor again. We did a second gate return and a new log book maintenance entry log page. No one needed medical assistance.All passengers were deplaned and an aircraft swap was planned; but needed to be recrewed due to flight deck and FAs exceeding CCO (Critical Crew Off) time.