  In this paper, we formulate and analyse a geometric low-regularity integrator
for solving the nonlinear Klein-Gordon equation in the $d$-dimensional space
with $d=1,2,3$. The integrator is constructed based on the two-step
trigonometric method and thus it has a simple form. Error estimates are
rigorously presented to show that the integrator can achieve second-order time
accuracy in the energy space under the regularity requirement in
$H^{1+\frac{d}{4}}\times H^{\frac{d}{4}}$. Moreover, the time symmetry of the
scheme ensures its good long-time energy conservation which is rigorously
proved by the technique of modulated Fourier expansions. A numerical test is
presented and the numerical results demonstrate the superiorities of the new
integrator over some existing methods.
