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 balanced rigid if for any precompact subset $B \subset\mathbb{R}^*$, the 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 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 $\mathbb{P}$. 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 Whit-taker kernels of Borodin and Olshanski are balanced rigid and have the balanced Palm equivalence property.