  The precise crystal symmetry and hence the emergence of the electric
polarization still remains an open question in the multiferroic materials
$R$Mn$_2$O$_5$ ($R$ = rare-earth, Bi, Y). While previous diffraction studies
have indicated that $R$Mn$_2$O$_5$ possesses the centro-symmetric space group
P$bam$, an atomic displacement allowing for the electric polarization would
require a non-centrosymmetric crystal symmetry. Our single crystal neutron
diffraction experiments on TbMn$_2$O$_5$ provide direct evidence of a reduced
crystallographic symmetry already above the magnetic and ferroelectric phase
transitions and a change in magnetic order upon entering the ferroelectric
phase. This is indicated through the presence of additional nuclear Bragg
reflections that are otherwise forbidden for the space group P$bam$ but are in
good agreement with the polar space group P$12_11$. It implies that the
exchange-striction, which arises from a symmetric $S_i \cdot S_j$ spin
coupling, is the dominating mechanism for the generation of the electric
polarization in the commensurate magnetic phase of TbMn$_2$O$_5$. Furthermore,
the commensurate magnetic reflections are in accordance with a quartile step
spin-spiral along the $c$-axis. Therefore, the antisymmetric $S_i \times S_j$
exchange via the inverse Dzyaloshinskii-Moriya interaction contributes as well
and becomes the leading term in the low temperature incommensurate spin-spiral
magnetic phase. These new findings provide important information for the
understanding of the complex interplay between the magnetic and the structural
order throughout the $R$Mn$_2$O$_5$ series of type-II multiferroics.
