It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via $\chi(F)=n(\chi(B)-\sum_i^r\frac{a_i-1}{a_i})$, where $r$ is the number of cones with stabilizer of orders $a_1,...,a_r$ respectively.

Now, I don't know if everyone would like to know what is the corresponding relation when $B$ has *reflector intervals* or *reflector circles*, as I'd rather... so I dare to question:

Is there a generalization in this direction?