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We say that something is symmetric, or symmetrical, if it has some kind of symmetry. In other words, there is some transformation we can perform on it which will leave it unchanged.
A shape is symmetrical if it is unchanged by a reflection, or a rotation.
A matrix is symmetric if it remains the same when we reflect it about its diagonal.

Of a relation  ∼  : x  ∼ y \implies y  ∼ x for all x,y in the domain of  ∼  .
Of a bilinear form ϕ: V² → F : satisfying ϕ(v,w) = ϕ(w,v) ∀v,w ∈ V .