Nonsymmetrical Orbits?

[cantab]

So, to sum it up, the 8th grade theory of seasons (the earth moves closest at summer, furthest at winter, equal at spring and summer) is not possible because of orbital mechanics.

On the contrary it's perfectly possible for that to be the case on other worlds, if the orbit is elliptical enough, the axial tilt is small enough, and the day short compared to the year. Seasons dominated by the shape of the orbit will be the same across the whole planet, whereas seasons dominated by the tilt of the spin axis are opposite in northern and southern hemispheres, and if the day is very long then the daily changes mask the yearly ones.

I don't think any large object in the solar system qualifies. Mars is a borderline case - while its seasons are dominated by axial tilt they're significantly affected by the distance from the Sun which also affects the speed of the planet in its orbit. Mars' southern hemisphere has a long cold winter with a short hot summer, while the northern hemisphere has a short mild winter and a long but cool summer.

[YNM]

Seasons are due to Earth's axial tilt wrt. its orbital plane, not due to the eccentricity of Earth's orbit. Just draw an ellipse with eccentricity of ~0.0167 and compare that with a circle. You won't be able to eyeball any difference !

On the question: If you consider apsidal precession (due to GR) and perturbation by other bodies, then there are no orbits which are repeatable and symmetric. For a while it might not make much difference, but on astronomical epochs it'd make sense, like changing a Sun - orbiting comet into Jupiter orbiting, or making a binary system falls closer to a black hole and component that survives become runaway stars. Or even the fact that Sun is wobbling in and out of the galactic plane, with no correlation to its galactic orbital period.