As you say, nature does love symmetry, but nature also loves to break symmetry. True symmetries in nature are tied with laws of conservation in physics and are beautiful things to look for because they usually say something deep about the structure of the universe. The symmetry of physical events in time (a particular event with the same initial conditions will take place according to the same physical rules at different times) implies the law of conservation of energy.
But symmetry of shapes is something else entirely. Planets tend to be formed in circular orbits simply because they form from debris disks surrounding their young parent stars, and material in the disk that has an elliptical component to its velocity quickly hits other material in the disk (or is slowed by the friction of interacting with gas in the disk) so that it ends up circularized. Clumps of extra density in the disk end up attracting more matter and planets slowly form and enlarge until the disk is essentially gone (the parent star also helps clean out the remnant material with the strong solar wind associated with young stars). But once the density of gas and dust in a disk is gone, the forces that tend toward circularization of orbits are also gone -- there is no longer any "viscous" force acting on particles which move outside of a circular path.
At this point, interactions among the planets themselves start disturbing the early symmetry. Planets whose orbital periods are near multiples of each other start tugging on each other repeatedly in the same place in the orbit, distorting their orbits towards an increasing ellipticity. Close encounters among planets whose orbits have evolved towards sufficient ellipticity that they may approach one another can lead to major rapid changes in an orbital shape -- and can even result in the ejection of a planet altogether. A collision may even occur.
If planets did not interact with each other, none of this would occur -- but gravity is a universal force, acting between all objects with mass. Planets tend to have a tiny amount of the mass of their parent star (in our solar system, the total mass of all planets is about a thousandth that of our Sun) but over time, these interactions can become profound.
So there are symmetrizing and anti-symmetrizing circumstances that occur in physics. In planet formation, both regimes occur -- planet formation occurs in a symmetrizing regime where frictional forces circularize orbits. Once these are gone, the interactions of planets -- an antisymmetrizing force -- dominate.
We're lucky the forces are small, otherwise our Solar System could have turned into a pinball machine a long time ago. It's mostly stable, but not entirely so. We cannot predict the specific position of the planets billions of years hence, because the tiniest error in our current knowledge becomes amplified to huge errors that far in the future, but we do know that in 2 billion years, there's a substantial chance that Mercury may get kicked into a very different orbit, and Uranus and Neptune may have close encounters which would profoundly change their orbits -- or even eject one of them.
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