Your quotient manifold is homeomorphic to $S^3$. I know this because it is a closed 3-manifold and its fundamental group is trivial, so I'm quoting the Poincare conjecture/Perelmann's theorem. The fundamental group can be seen to be trivial because there is a CW-decomposition of the (2-manifold) torus with one 0-cell, three 1-cells and two 2-cells that is respected by your involution, where the 1-cells are the longitude, meridian and diagonal of the torus, and the two 2-cells are triangles. The involution swaps the longitude and meridian while fixing the diagonal pointwise, and swaps the two 2-cells. The quotient of the torus by the involution has one vertex, two 1-cells and one 2-cell. The fundamental group of this space is infinite cyclic, generated by the loop around the 1-cell that started life as both the longitude and meridian of the torus.
This quotient space can be viewed as a copy of the Mobius strip, as in Taras Banakh's comment.
But now think about how the insides fits in: a CW-structure on the solid torus can be made by adding in one more 2-cell, attached around the meridian of the torus, and one 3-cell. The extra 2-cell wraps around the generator for the fundamental group of the piece discussed in the previous paragraph. So the fundamental group of the overall space is trivial.