In this paper we present a method of obtaining new examples of spaces of orderings by considering quotient structures of the space of orderings $(X_{\mathbb{Q}\left(x\right)},G_{\mathbb{Q}\left(x\right)})$ - it is, in general, nontrivial to determine whether, for a subgroup $G₀\subset G_{\mathbb{Q}\left(x\right)}$ the derived quotient structure $(X_{\mathbb{Q}\left(x\right)}{|}_{G₀},G₀)$ is a space of orderings, and we provide some insights into this problem. In particular, we show that if a quotient structure arising from a subgroup of index 2 is a space of orderings, then it necessarily is a profinite one.