Do planetary orbits rotate?

Yes. The orbits are ellipses, with one focus occupied by the sun. The location in space of the perihelion of each planet shifts along a circle whose radius is the parahelion distance, by an amount that depends on how close the planet is to the sun. The precession rates, in seconds of arc per century, are as follows:

Planet                Total rate   Part due to general relativity   
Mercury...............5599                   43.0
Venus.................5069                    8.6
Earth.................6189                    3.84
Mars..................6627                    1.35
Jupiter...............5800                    0.06
Saturn................7053
Uranus................5846
Neptune...............3161
Now, these numbers all include a contribution of 5000 arcseconds per century which is the 'precession of the equinoxes' of the Earth which causes our sky coordinate system to shift and has nothing to do with the actual perihelion shift itself. This should be subtracted from each of the numbers above.
Copyright 1997 Dr. Sten Odenwald
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