Brain teaser! Symmetry implies a uniform arrangement with two sides being similar, but before we could identify any symmetry we would need to know the shape and boundaries (if the universe had a shape or boundaries) and all that lay within.
I would have to say the universe is not symmetrical, however the possibility of another milky way (or a band of close proximity stars, planets and solar systems) is likely.
There are other solar systems out there, according to NASA, there are a number of stars, alive and dying, that have planets circling them.
As for Earth, there is a strong possibility of a planet similar in size and at the right distance from its sun that is home to animal and plant life forms.
There is even a stronger possibility of many planets that contain plant and animal life forms. Unfortunately our scientists only seem to look for ones similar to ours that could sustain life in the forms we have on earth.
Considering the diversity of both animal and plant life on earth, one needs to expand the mind to consider the possibility of life being able to withstand grater or less gravitational force, to breath in an atmosphere that is not predominantly Nitrogen and Oxygen or to survive in extreme temperatures.