Summary: Following Riley’s work, for each 2-bridge link \(K(r)\) of slope \(r \in \mathbb Q\) and an integer or a half-integer \(n\) greater than 1, we introduce the Heckoid orbifold \(\mathbf S(r; n)\) and the Heckoid group \(G(r; n)=\pi_1(\mathbf S(r; n))\) of index \(n\) for \(K(r)\). When \(n\) is an integer, \(\mathbf S(r; n)\) is called an even Heckoid orbifold; in this case, the underlying space is the exterior of \(K(r)\), and the singular set is the lower tunnel of \(K(r)\) with index \(n\). The main purpose of this note is to announce answers to the following questions for even Heckoid orbifolds. (1) For an essential simple loop on a 4-punctured sphere \(\mathbf S\) in \(\mathbf S(r; n)\) determined by the 2-bridge sphere of \(K(r)\), when is it null-homotopic in \(\mathbf S(r; n)\)? (2) For two distinct essential simple loops on \(\mathbf S\), when are they homotopic in \(\mathbf S(r; n)\)? We also announce applications of these results to character varieties, McShane’s identity, and epimorphisms from 2-bridge link groups onto Heckoid groups.