To get a parametrization, take any standard 2-D figure 8 curve, such
as a lemniscate. Starting in the x-z plane, rotate it around the
z-axis, getting a torus, but rotate your lemniscate theta/2, where
theta is the angle of rotation about the z-axis.  The figure 8's will
match up when they come back together, but one will be rotated 180
degrees with respect to the other.

  Another way to view it is as a mobius strip with the top and bottom
halves joined.

  If you're really into warping your brain, try viewing the real
projective plane.  This is a hemisphere with diametrically opposed
points on the equator identified.  You can view this as a sort of
figure 8 bowl (the rim of the bowl looks like a figure 8), with the
two lobes of the 8 folded up to touch each other.  In 4 dimensions you
can again twiddle with the parameters to avoid self intersection.