at.algebraic topology - classification of smooth involutions of torus

Let $mathbb{Z}_2={1,g},T^2={(e^{itheta_1},e^{itheta_2})}$ and place $T^2$ in $mathbb{R}^3$ as the locus of the rotation of $2pi$ rads of the circle${(y,z)|(y-2)^2+z^2=1}$ around $z$ axis.

It is known that there are 5 nonequivalent smooth involutions on torus,and they are:

1.$g(e^{itheta_1},e^{itheta_2})=(e^{i(theta_1+pi)},e^{itheta_2})$ (rotation$pi$ rads around $z$ axis) with null fixed point set and orbit space $T^2$

2.$g(e^{itheta_1},e^{itheta_2})=(e^{-itheta_1},e^{itheta_2})$(reflection along $x=0$) with fixed point set $S^1times S^0$ and orbit space an annulus

3.$g(e^{itheta_1},e^{itheta_2})=(e^{itheta_2},e^{itheta_1})$(switch the two coordinates) with fixed point set the diagonal circle and orbit space Mobius band

4.$g(e^{itheta_1},e^{itheta_2})=(e^{i(theta_1 +pi)},e^{-itheta_2})$(restriction of the involution $(x,y,z,mapsto (-x,-y,-z)$ of $mathbb{R}^3$ to torus)with null fixed point set and orbit space klein bottle

5.$g(e^{itheta_1},e^{itheta_2})=(e^{-itheta_1},e^{-itheta_2})$(reflection along $x=0$ plus reflection along $z=0$)
with fixed point set 4 points and orbit space $S^2$

i want to know how to derive the result above.for the free case it seems easy.since the action is free,the orbit space must be a manifold also,and has euler char 0,hence must be torus or klein bottle.
for the nonfree case,the orbit is not manifold,but "orbifold".
and we have Riemann-Hurwitz Formula:

$chi(O)=chi(X_O)-sum_{i=1}^n (1-frac{1}{q_i})-frac{1}{2}sum_{j=1}^m (1-frac{1}{r_j})$

here$chi(O)$ is the orbifold euler char and $chi(X_o)$ is the euler char of the underlying space associated to the orbifold $O$,and $q_i$and $r_j$ denote the angles for sigular points(cone points and reflector corners
can we determine the remaining 3 involutions by using this formula?Thank you!