We characterize smooth functions on R+d that admit η-symmetric smooth extensions onto Rd. Also, we characterize smooth functions on R+d that are restrictions of η-symmetric Schwartz class functions on Rd. Here η∈Z2d={0,1}d and η-symmetry of f:Rd→C means that f is either even or odd with respect to the ith coordinate according to whether ηi=0 or ηi=1, respectively. As an application we show that the multi-dimensional Hankel transforms conjugated by a natural multiplier operator have the restrictions to R+d of η-symmetric Schwartz class functions on Rd as their test function spaces.