  We define an analogue of the Arnold surface for odd degree flexible curves,
and we use it to double branch cover $Q$-flexible embeddings, where
$Q$-flexible is a condition to be added to the classical notion of a flexible
curve. This allows us to obtain a Viro--Zvonilov-type inequality: an upper
bound on the number of non-empty ovals of a curve of odd degree. We investigate
our method for flexible curves in a quadric to derive a similar bound in two
cases. We also digress around a possible definition of non-orientable flexible
curves, for which our method still works and a similar inequality holds.
