No. Just as the Directv satellites use right-hand and left-hand circular, many other satellites use vertical and horizontal polarization. The transmitting antennas are different (circular polarization has both vertical and horizontal components).
I guess given the additional LNB skew adjustment complexities discussed here, the question should be asked why communications satellite designers past and present ever chose linear polarization over circular to begin with?
I was even taught way back in my college days that a linearly polarized satellite signal or other microwave transmission can even experience an affect called "Faraday Rotation," when transiting the ionosphere to or from space, where in severe cases can cause a signal originally transmitted as horizontal to be received vertically.