  The solutions that describe the motion of the classical simple pendulum have
been known for very long time and are given in terms of elliptic functions,
which are doubly periodic functions in the complex plane. The independent
variable of the solutions is time and it can be considered either as a real
variable or as a purely imaginary one, which introduces a rich symmetry
structure in the space of solutions. When solutions are written in terms of the
Jacobi elliptic functions the symmetry is codified in the functional form of
its modulus, and is described mathematically by the six dimensional coset group
$\Gamma/\Gamma(2)$ where $\Gamma$ is the modular group and $\Gamma(2)$ is its
congruence subgroup of second level. In this paper we discuss the physical
consequences this symmetry has on the pendulum motions and it is argued they
have similar properties to the ones termed as duality symmetries in other areas
of physics, such as field theory and string theory. In particular a single
solution of pure imaginary time for all allowed value of the total mechanical
energy is given and obtained as the $S$-dual of a single solution of real time,
where $S$ stands for the $S$ generator of the modular group.
