  We report the occurrence of both \emph{ab} and \emph{bc} cycloidal ordering
of Mn-spins at different temperatures and their possible coexistence at low
temperatures in the polycrystalline mixed rare-earth compounds,
\emph{R}$_{0.5}$Dy$_{0.5}$MnO$_3$ (\emph{R} = Eu and Gd), which exhibit
extraordinary magnetoelectric properties. While the polarization of
Gd$_{0.5}$Dy$_{0.5}$MnO$_3$ is comparable to TbMnO$_3$, the compound
Eu$_{0.5}$Dy$_{0.5}$MnO$_3$ shows high value of polarization. However, both of
them show giant magnetic tunability and exhibit large magnetocapacitance whose
sign changes across the two cycloidal ordering temperatures. Intriguingly, the
electric polarization can be reversed upon ramping up or ramping down the
magnetic field, which has not been observed for any of the \emph{R}MnO$_3$
system. Most strikingly, these compounds show non-volatile ferroelectric memory
effect even in the paraelectric and paramagnetic region (T$_C$ $\leq$ T $\leq$
80 K). We attribute these remarkable properties to the coexistence of \emph{ab}
and \emph{bc} cycloidal ordered phases.
