'Sym' usually means 'like' or 'same' and sometimes 'with.' Consider the following examples.
If something is symmetrical (having symmetry), it is the same form on all sides; they are identical. The Taj Mahal would be an architectural example of this. A Rorschach ink blot formed by folding a paper in half and then pressing it is an example of a symmetrical pattern. Sometimes the prefix is written 'sim' as in 'simultaneous,' meaning 'happening at the same time.'
To feel sympathy, one feels compassion (feels with) someone else's situation without necessarily being under the same circumstances .(That would be 'empathy.' ) The word 'symphony' means the state or condition of being (-y) like or compatible (sym-) sounds (phon).