  Let $A$ and $A'$ be abelian varieties defined over a number field $k$ of
dimension $g\geq 1$. For $g\leq 3$, we show that the following local-global
principle holds: $A$ and $A'$ are quadratic twists of each other if and only
if, for almost all primes $\mathfrak p$ of $k$ of good reduction for $A$ and
$A'$, the reductions $A_{\mathfrak p}$ and $A'_{\mathfrak p}$ are quadratic
twists of each other. This result is known when $g=1$, in which case it has
appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an
example that violates this local-global principle in dimension $g=4$.
