We generalize the concept of the $k$-th symmetric difference in the sense of Stein and Zygmund to that of symmetric difference with respect to a weight system of order $n$ and the concept of symmetrically continuous functions and symmetric functions to that of functions symmetric with respect to a weight system of order $n$. We also study the classes of even symmetry and odd symmetry consisting of functions whose limits to the right and to the left exist at each point; hence, their set of points of discontinuity is countable, and they are in Baire class one. The functions symmetric with respect to a fixed weight system $W_n$ of order $n$ form a linear space $V(W_n)$, and the subclass $B(W_n)$ consisting of bounded functions forms a Banach space with the norm $\|f\|=\sup |f(x)|$.