In this paper we consider twist mappings of the torus, $\overline{T}:{\rm T^2\rightarrow T^2,}$ and their vertical rotation intervals $\rho _V( T)=[\rho _V^{-},\rho _V^{+}],$ which are closed intervals such that for any $\omega \in ]\rho _V^{-},\rho _V^{+}[$ there exists a compact $ \overline{T}$-invariant set $\overline{Q}_\omega $ with $\rho _V(\overline{x}% )=\omega$ for any $\overline{x}\in \overline{Q}_\omega ,$ where $\rho _V( \overline{x})$ is the vertical rotation number of $\overline{x}.$ In case $% \omega $ is a rational number, $\overline{Q}_\omega $ is a periodic orbit (this study began in [1] and [2]). Here we analyze how $\rho _V^{-}$ and $% \rho _V^{+}$ behave as we perturb $\overline{T}$ when they assume rational values. In particular we prove that, for an interesting class of mappings, these functions are locally constant at rational values.