  Isometric class of minimal surfaces in the Euclidean 3-space $\mathbb{R}^3$
has the rigidity: if two simply connected minimal surfaces are isometric, then
one of them is congruent to a surface in the specific one-parameter family,
called the associated family, of the other. On the other hand, the situation
for surfaces with Lorentzian metrics is different. In this paper, we show that
there exist two timelike minimal surfaces in the Lorentz-Minkowski 3-space
$\mathbb{R}^3_1$ that are isometric each other but one of which does not belong
to the congruent class of the associated family of the other. We also prove a
rigidity theorem for isometric and anti-isometric classes of timelike minimal
surfaces under the assumption that surfaces have no flat points. Moreover, we
show how symmetries of such surfaces propagate for various deformations
including isometric and anti-isometric deformations. In particular, some
conservation laws of symmetry for Goursat transformations are discussed.
