  In this paper and its sequel, we give a necessary and sufficient condition
for two essential simple loops on a 2-bridge sphere in an even Heckoid orbifold
for a 2-bridge link to be homotopic in the orbifold. We also give a necessary
and sufficient condition for an essential simple loop on a 2-bridge sphere in
an even Heckoid orbifold for a 2-bridge link to be peripheral or torsion in the
orbifold. This paper treats the case when the 2-bridge link is a $(2,p)$-torus
link, and its sequel will treat the remaining cases.
