Even functions are symmetric with respect to the y-axis, meaning their graph remains unchanged when x is replaced by -x, while odd functions are symmetric with respect to the origin, meaning their graph remains unchanged when both x and y are replaced by their negatives. These symmetries lead to specific algebraic properties: for an even function, f(x) = f(-x) for all x in the domain, and for an odd function, f(-x) = -f(x) for all x in the domain.