Semimetallic tungsten ditelluride displays an extremely large nonsaturating magnetoresistance, which is thought to arise from the perfect $n\ensuremath{-}p$ charge compensation with low carrier densities in $\mathrm{WT}{\mathrm{e}}_{2}$. We find a strong Rashba spin-orbit effect in density functional calculations due to the noncentrosymmetric structure. This lifts twofold spin degeneracy of the bands. A prominent umklapp interference pattern is observed by our scanning tunneling microscopic measurements at 4.2 K, which differs distinctly from the surface atomic structure demonstrated at 77 K. The energy dependence of umklapp interference shows a strong correspondence with densities of states integrated from ARPES measurement, manifesting a fact that the bands are spin-split on the opposite sides of \ensuremath{\Gamma}. Spectroscopic survey reveals the electron/hole asymmetry changes alternately with lateral locations along the $b$ axis, providing a microscopic picture for double-carrier transport of semimetallic $\mathrm{WT}{\mathrm{e}}_{2}$. The conclusion is further supported by our ARPES results and Shubnikov--de Haas (SdH) oscillations measurements.