  Coherent optical spectroscopy such as four-wave mixing and photon echo
generation deliver detailed information on the energy levels involved in
optical transitions through the analysis of polarization of the coherent
response. In semiconductors, it can be applied to distinguish between different
exciton complexes, which is a highly non-trivial problem in optical
spectroscopy. We develop a simple approach based on photon echo polarimetry, in
which polar plots of the photon echo amplitude are measured as function of the
angle $\varphi$ between the linear polarizations of the two exciting pulses.
The rosette-like polar plots reveal a distinct difference between the neutral
and charged exciton (trion) optical transitions in semiconductor
nanostructures. We demonstrate this experimentally by photon echo polarimetry
of a 20-nm-thick CdTe/(Cd,Mg)Te quantum well at temperature of 1.5~K. Applying
narrow-band optical excitation we selectively excite different exciton
complexes including the exciton, the trion, and the donor-bound exciton D$^0$X.
We find that polarimetry of the photon echo on the trion and D$^0$X is
substantially different from the exciton: The echoes of the trion and D$^0$X
are linearly polarized at the angle $2\varphi$ with respect to the first pulse
polarization and their amplitudes are weakly dependent on $\varphi$. While on
the exciton the photon echo is co-polarized with the second exciting pulse and
its amplitude scales as $\cos\varphi$.
