It is known that some traditional shock-capturing schemes, including Godunov- or Roe-type schemes, generate downstream oscillatory wave patterns that cannot be effectively dampened by the dissipation methods they provide. The theme of this paper is to understand the formation and behavior at these upstream/downstream patterns from the point view of the symmetry inherent in the PDE. This paper investigates symmetry breaking and spatial oscillations in finite difference solutions for the inviscid Burgers and one-dimensional Navier-Stokes shock equations using the Lax-Wendroff, TVD, and nonoscillatory, nonfree parameter dissipative schemes. Symmetry breaking is considered as a candidate for the inherent cause of the nonphysical oscillations in the vicinity of the shock.