Active microswimmers are known to affect the macroscopic viscosity of suspensions in a more complex manner than passive particles. For puller-like microswimmers an increase in the viscosity has been observed. It has been suggested that the persistence of the orientation of the microswimmers hinders the rotation that is normally caused by the vorticity. It was previously shown that some sorts of algae are bottom-heavy swimmers, i.e., their centre of mass is not located in the centre of the body. In this way, the algae affect the vorticity of the flow when they are perpendicularly oriented to the axis of gravity. This orientation of gravity to vorticity is given in a rheometer that is equipped with a cone-plate geometry. Here we present measurements of the viscosity both in a cone-plate and a Taylor-Couette cell. The two set-ups yielded the same increase in viscosity although the axis of gravitation in the Taylor-Couette cell is parallel to the direction of vorticity. In a complementary experiment we tested the orientation of the direction of swimming through microscopic observation of single Chlamydomonas reinhardtii and could not identify a preferred orientation, i.e., our specific strain of Chlamydomonas reinhardtii are not bottom-heavy swimmers. We thus conclude that bottom heaviness is not a prerequisite for the increase of viscosity and that the effect of gravity on the rheology of our strain of Chlamydomonas reinhardtii is negligible. This finding reopens the question of whether the origin of persistence in the orientation of cells is actually responsible for the increased viscosity of the suspension.