  Is the recent evidence for a time-varying fine structure 'constant' $\alpha$
to be interpreted as a varying $e$, $c$, $\hbar$, or a combination thereof? We
consider the simplest varying electric charge and varying speed of light
theories (VSL) and prove that for the same type of dark matter they predict
opposite senses of variation in $\alpha$ over cosmological times. We also show
that unlike varying $e$ theories, VSL theories do not predict violations of the
weak equivalence principle (WEP). Varying $e$ theories which explain
astronomical inferences of varying $\alpha$ predict WEP violations only an
order of magnitude smaller than existing E\"otv\"os experiment limits but could
be decisively tested by STEP. We finally exhibit a set of atomic-clock and
related experiments for which {\it all} (hyperbolic) varying $\alpha$ theories
predict non-null results. They provide independent tests of the recent
astronomical evidence.
