We determine the inclinations of the vector magnetic field and flow velocity in a sunspot penumbra by interpreting full Stokes profiles of three infrared lines observed with the Tenerife Infrared Polarimeter. It is shown that analyses based on one-component atmospheres deliver flow velocities which are more horizontal than the average magnetic field by up to 10 deg. This apparent violation of the concept of frozen-in magnetic fields is solved as soon as two magnetic atmospheres are allowed to coexist in the resolution element. The magnetic field and velocity in the atmospheric component carrying the Evershed flow are found to be aligned to within +/- 2 deg all the way from the inner to the outer penumbra. This is the first observational confirmation of magnetic fields being frozen into the plasma in sunspots. Our results indicate that sunspot penumbrae can be understood in terms of inclined flux tubes embedded in a more vertical background field. The flux tubes carry most of the Evershed flows and return to the solar surface in the middle penumbra and beyond. The background atmosphere is essentially at rest in the inner penumbra, and harbors small flows in the outer penumbra.