  We study Palm measures of determinantal point processes with $J$-Hermitian
correlation kernels. A point process $\mathbb{P}$ on the punctured real line
$\mathbb{R}^* = \mathbb{R}_{+} \sqcup \mathbb{R}_{-}$ is said to be
$\textit{balanced rigid}$ if for any precompact subset $B\subset \mathbb{R}^*$,
the $\textit{difference}$ between the numbers of particles of a configuration
inside $B\cap \mathbb{R}_{+}$ and $B\cap \mathbb{R}_{-}$ is almost surely
determined by the configuration outside $B$. The point process $\mathbb{P}$ is
said to have the $\textit{balanced Palm equivalence property}$ if any reduced
Palm measure conditioned at $2n$ distinct points, $n$ in $\mathbb{R}_{+}$ and
$n$ in $\mathbb{R}_{-}$, is equivalent to the $\mathbb{P}$.
  We formulate general criteria for determinantal point processes with
$J$-Hermitian correlation kernels to be balanced rigid and to have the balanced
Palm equivalence property and prove, in particular, that the determinantal
point processes with Whittaker kernels of Borodin and Olshanski are balanced
rigid and have the balanced Palm equivalence property.
