  Neutron diffraction and muon spin relaxation ($\mu$SR) studies are presented
for the newly characterized polymorph of NiNb$_2$O$_6$ ($\beta$-NiNb$_2$O$_6$)
with space group P4$_2$/n and $\mu$SR data only for the previously known
columbite structure polymorph with space group Pbcn. The magnetic structure of
the P4$_2$/n form was determined from neutron diffraction using both powder and
single crystal data. Powder neutron diffraction determined an ordering wave
vector $\vec{k}$ = ($\frac{1}{2},\frac{1}{2},\frac{1}{2}$). Single crystal data
confirmed the same $\vec{k}$-vector and showed that the correct magnetic
structure consists of antiferromagnetically-coupled chains running along the a
or b-axes in adjacent Ni$^{2+}$ layers perpendicular to the c-axis, which is
consistent with the expected exchange interaction hierarchy in this system. The
refined magnetic structure is compared with the known magnetic structures of
the closely related tri-rutile phases, NiSb$_2$O$_6$ and NiTa$_2$O$_6$. $\mu$SR
data finds a transition temperature of $T_N \sim$ 15 K for this system, while
the columbite polymorph exhibits a lower $T_N =$ 5.7(3) K. Our $\mu$SR
measurements also allowed us to estimate the critical exponent of the order
parameter $\beta$ for each polymorph. We found $\beta =$ 0.25(3) and 0.16(2)
for the $\beta$ and columbite polymorphs respectively. The single crystal
neutron scattering data gives a value for the critical exponent $\beta
=$~0.28(3) for $\beta$-NiNb$_2$O$_6$, in agreement with the $\mu$SR value.
While both systems have $\beta$ values less than 0.3, which is indicative of
reduced dimensionality, this effect appears to be much stronger for the
columbite system. In other words, although both systems appear to
well-described by $S = 1$ spin chains, the interchain interactions in the
$\beta$-polymorph are likely much larger.
