A rank-two tensor is going to have 3 principle axes that can be visualized. You will end up with 3 axes and the best way to visualize these at a point is with an oblate spheroid at each point. Effectively you just rotate the ellipse created by 2 of the principle axes about the third. Various software packages can do this, I prefer ParaView but to each their own.
Higher order than that though and you will not have a very good way to visualize it. I would prefer to move to a different type of analysis and instead begin to look at the topology of the field instead and study the tensor invariants. These invariants are the coefficients of the characteristic equation, which for rank-two is:
$$ \lambda^3 + P \lambda^2 + Q \lambda + R = 0 $$
Similar expressions exist of course for higher dimensions. This paper and this paper provide algorithms for computing the invariants, and provide the expressions for a rank-four tensor. In the context of my work, fluid dynamics, this is done with the velocity gradient tensor, or the rate-of-strain tensor or the rate-of-rotation tensor. The $(P,Q,R)$ space is divided by the discriminant surfaces, and these can be assigned to topological features. In the rank-two example, there are 8 sectors that correspond to focuses, saddles, nodes etc that may be stable or unstable (see for example this paper for applications in fluids). Again, drawing back to my work, these can be assigned physical properties. For example, unstable focus/compressing corresponds to vortex compression. Unstable node/saddle/saddle is a vortex sheet while stable node/saddle/saddle is a vortex tube. I'm sure other descriptions could be attributed in your case, and for higher order invariants. The invariants themselves also may have physical meaning. For fluids, $P$ is the volumetric expansion/compression and $Q$ is related to the rotation.
The last topological technique I am familiar with is the Morese-Small Complex. In this, you take a field and identify the critical points -- local minima, maxima, saddles, and nodes. These points are then connected together through the field and the boundaries around each critical point identify the flow of information along the topology. It is useful for creating topological maps of high dimensional datasets.