All contributing magnetic anisotropies in (110)-oriented exchange biased Ni80Fe20/Fe50Mn50 double layers prepared by molecular beam epitaxy on Cu(110) single crystals have been determined by means of Brillouin light scattering. Upon covering the Ni80Fe20 films by Fe50Mn50, a unidirectional anisotropy contribution appears, which is consistent with the measured exchange bias field. The uniaxial and fourfold in-plane anisotropy contributions are largely modified by an amount, which scales with the Ni80Fe20 thickness, indicating an interface effect. The strong uniaxial anisotropy contribution shows an in-plane switching of the easy axis from  to  with increasing Ni80Fe20-layer thickness. The large mode width of the spin wave excitations, which exceeds the linewidth of uncovered Ni80Fe20 films by a factor of more than six, indicates large spatial variations of the exchange coupling constant.