For free Klein-Gordon fields, we construct a one-parameter family of
conserved current densities $J_a^\mu$, with $a\in(-1,1)$, and use the latter to
yield a manifestly covariant expression for the most general positive-definite
and Lorentz-invariant inner product on the space of solutions of the
Klein-Gordon equation. Employing a recently developed method of constructing
the Hilbert space and observables for Klein-Gordon fields, we then obtain the
probability current density ${\cal J}_a^\mu$ for the localization of a
Klein-Gordon field in space. We show that in the nonrelativistic limit both
$J_a^\mu$ and ${\cal J}_a^\mu$ tend to the probability current density for the
localization of a nonrelativistic free particle in space, but that unlike
$J_a^\mu$ the current density ${\cal J}_a^\mu$ is neither covariant nor
conserved. Because the total probability may be obtained by integrating either
of these two current densities over the whole space, the conservation of the
total probability may be viewed as a consequence of the local conservation of
$J_a^\mu$. The latter is a manifestation of a previously unnoticed global gauge
symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if
the parameter $a$ is rational. It is the multiplicative group of positive real
numbers if $a$ is irrational. We also discuss an extension of our results to
Klein-Gordon fields minimally coupled to an electromagnetic field.