The authors study the qualitative behavior of entropy solutions to the two-dimensional Riemann problem for a single conservation law, i.e., they consider the equation \[ \partial u/\partial t+\partial f(u)/\partial x+\partial g(u)/\partial y=0 \] with initial data constant in each quadrant of the plane. They construct solutions within the class of piecewise smooth solutions for the transformed equation obtained via the similarity transformation \(\xi =x_ t\), \(\eta =y/t\). Under the assumption \(f''g''(f''/g'')'\neq 0\) it is shown that continuous solutions take at most three forms which, transformed back into the (t,x,y)-plane, give rise to constant states, centered planar waves and centered wave cones. The analysis is done thoroughly and presented in full detail.