On the symmetry of a $2$-dimensional discrete-time signal
What you have displayed is true for which it's odd symmetric for only one of the variables $n1,n2$ and even-symmetric for th other.

For 1D signals, an even-symmetric signal has the property that
$$ h[-n] = h[n] $$

and an odd-symmetric signal has the property as:
$$ h[-n] = - h[n] $$

For 2D signals, like $h[n_1,n_2]$, more symmetries are possible:

For even signals, the signal with the property:
$$ h[-n_1,n_2] = h[n_1,n_2]$$
is even-symmetric only in $n_1$ variable, whereas the signal with the property 
$$ h[n_1,-n_2] = h[n_1,-n_2]$$
is even-symmetric only in $n_2$ variable. Consequently the signal withe property that
$$ h[-n_1,-n_2] = h[n_1,-n_2] = h[-n_1,n_2] = h[n_1,n_2]$$
is even-symmetric in both $n_1$ and $n_2$.

For odd signals, the signal with the property:
$$ h[-n_1,n_2] = -h[n_1,n_2]$$
is odd-symmetric only in $n_1$ variable, whereas the signal with the property 
$$ h[n_1,-n_2] = -h[n_1,n_2]$$
is even-symmetric in $n_2$ variable. Consequently
$$ h[-n_1,-n_2] = -h[n_1,-n_2] = -h[-n_1,n_2] = h[n_1,n_2]$$
is odd-symmetric in both $n_1$ and $n_2$.

So the property you have displayed 
$$ h[-n_1,-n_2] = -h[n_1,n_2]$$

implies an even-symmetry on $n_1$ and odd-symmetry on $n_2$ OR an odd-symmetry on $n_1$ and even-symmetry on $n_2$