  Isotropic and anisotropic magnetic behavior of the frustrated spin chain
compound $\beta$-TeVO$_4$ is reported. Three magnetic transitions observed in
zero magnetic field are tracked in fields applied along different
crystallographic directions using magnetization, heat capacity, and
magnetostriction measurements. Qualitatively different temperature-field
diagrams are obtained below 10 T for the field applied along $a$ or $b$ and
along $c$, respectively. In contrast, a nearly isotropic high-field phase
emerges above 18 T and persists up to the saturation that occurs around 22.5 T.
Upon cooling in low fields, the transitions at $T_{\rm N1}$ and $T_{\rm N2}$
toward the spin-density-wave and stripe phases are of the second order, whereas
the transition at $T_{\rm N3}$ toward the helical state is of the first order
and entails a lattice component. Our microscopic analysis identifies frustrated
$J_1-J_2$ spin chains with a sizable antiferromagnetic interchain coupling in
the $bc$ plane and ferromagnetic couplings along the $a$ direction. The
competition between these ferromagnetic interchain couplings and the helical
order within the chain underlies the incommensurate order along the
$a$-direction, as observed experimentally. Although a helical state is
triggered by the competition between $J_1$ and $J_2$ within the chain, the
plane of the helix is not uniquely defined because of competing magnetic
anisotropies. Using high-resolution synchrotron diffraction and $^{125}$Te
nuclear magnetic resonance, we also demonstrate that the crystal structure of
$\beta$-TeVO$_4$ does not change down to 10 K, and the orbital state of
V$^{4+}$ is preserved.
