  In this paper, we collect a number of facts about double Hurwitz numbers,
where the simple branch points are replaced by their more general analogues ---
completed (r+1)-cycles. In particular, we give a geometric interpretation of
these generalised Hurwitz numbers and derive a cut-and-join operator for
completed (r+1)-cycles. We also prove a strong piecewise polynomiality property
in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural
ELSV/GJV-type formula, that is, an expression in terms of some intrinsic
combinatorial constants that might be related to the intersection theory of
some analogues of the moduli space of curves. The structure of these
conjectural "intersection numbers" is discussed in detail.
