We show how the nonstandard $q$-deformation $U_qso_n$ can be used to decompose the $n$-fold tensor power of a spinor representation of the Drinfeld-Jimbo quantum group $U_qso_N$.As a second application,we use $U_qso_n$ and its centralizer on certain tensor powers of vector representations to construct categorical analogs of symmetric spaces for certain fusion tensor categories.