I am working on sinuous antenna, My question is that whether single sinuous antenna can be used for dual CP purpose simultaneously i.e. if it is transmitting LHCP wave and Receiving RHCP wave at the same time?
A single sinuous antenna can transmit/receive linear polarization only. Orientation of linear polarization depends on frequency. It wobbles about the main direction as the operation frequency changes but always remains linear. That wobbling causes confusion. It has nothing to do with circular polarization.
Vladimir F Shevchenko is right. It will receive both circular polarizations though, because linear polarization can be decomposed into RHC and LHC polarization. It might not pick them both up at once, because the sum might be the orthogonal linear polarization (it depends on the phase between the LHC and RHC), and each will have at least 3 dB loss, because of the polarization mismatch between linear and circular. As a linearly polarized antenna, it also transmits LHC and RHC in equal amounts at the same time.
Just small addition to the previous useful discussions
If you really want to have a dual-circular polarized sinuous antenna you can use the standard 4-elements sinuous antenna with the addition of a beamforming network composed of baluns and quadrature coupler to transform from dual-linear polarization to dual-circular one.
or a simpler solution, you can use spiral antennas which can directly provide you with dual circular polarization.
Attached are two papers for dual circular polarized antennas (both sinuous and spiral).