Universe Today
First six years (approx. 2001-2007): The orbit was mostly at 5 p.m. local solar time (as it flew north to south) and 5 a.m. local solar time on the south-to-north orbit. "That orbit provided an advantage for the orbiter's Gamma Ray Spectrometer to have its cooling equipment pointed away from the sun," NASA stated. At that time, the spectrometer found evidence of water ice, through the spectrum of hydrogen.
Next five years (approx. 2007-2012): The orbit shifted to 4 p.m. local solar time on north-to-south, and 4 a.m. south to north. While this allowed the Thermal Emission Imaging System to examine warm ground that made the mineral signatures in infrared pop out more easily, on the flip side of the planet Odyssey's power system was under more strain because the solar panels couldn't work as well in predawn light. Odyssey remained in that orbit until about the 2012 landing of the Curiosity rover, then was sent on a maneuver to move its orbit to later in the day to keep the battery functioning.
What's next: Once Odyssey is in the right spot, the spacecraft will flip its daylight observations to scan the ground at 6:45 a.m. on the south-to-north part of the orbit. The spacecraft was in fact going in that direction already, but the new maneuver gets it there a bit sooner.