It seems to me that whenever there is some material parameter for a continuum, it is described by a symmetric tensor. This is the case for the mechanical stress tensor, permittivity of crystals, the inertia tensor, even higher-order nonlinear susceptibility represented by third- and higher order tensors.
On the other hand, I don't know any material properties described by general, nonsymmetric tensors.
Is there an underlying reason for this that is shared by the examples provided?
This is a difficult subject and is a controversial one in thermodynamics. In 1851 Stokes noticed that in the generalized linear heat conduction written as $\mathbf q = -\mathbf K \cdot \nabla T,$ if the conductivity tensor $\mathbf K$ is not symmetrical then it would lead to a "... rotatory sort of motion of heat, produced by the mere diffusion from the source outwards, certainly seems very strange, and leads us to think" that $\mathbf K = \mathbf K^T$ always, and he managed to prove the symmetry for seven crystalline classes out of 13. Voigt, Curie, Soret tried to prove the purported symmetry for several crystal classes experimentally. For details, see Truesdell: Rational Thermodynamics, Chapter 7. and this post.
Truesdell also discusses the symmetry of the diffusion tensor in Fick's law for which Stefan proved for two component mixtures its symmetry based on the symmetry of action-reaction principle but not for ternary or more complex mixtures.
A symmetry result holds for linear viscosity defined by $\mathbf T =\mathbf V \nabla \dot {\mathbf x}$ where the $\mathbf V$ viscosity operator is assumed to have the form as $\mathbf V = \mathbf V(\mathbf {x, \dot x},t)$. Then $\mathbf V $ is a fourth order tensor between the stress $\mathbf T$ and the velocity gradient $\nabla \dot {\mathbf x}$ and it can be proved that it also reduces to a regular 2nd order one that is usually written as
$$\mathbf T = -p \mathbf I+\mathbf V \\
\mathbf V = \lambda \mathrm {tr}\{ \mathbf D\} \mathbf I + 2 \mu \mathbf D$$ with $2\mathbf D = \nabla \mathbf {\dot x} + \nabla \mathbf {\dot x}^T$ and consequently $\mathbf V$ is symmetrical.
In electromagnetism the permittivity tensor is indeed symmetrical but not the permeability tensor, say for example that of ferrites (see the Polder tensor) that are often used in microwaves and optics. That it is not a symmetrical tensor is the basis of all the nonreciprocal behavior widely exploited in applications, such as isolators, circulators and switches.
After Onsager's paper on microscopic reversibility in 1931 there was indeed a kind of "movement" to associate the symmetry of many constitutive tensors to this microscopic reciprocity. Maxwell's equations are not invariant to time reversal unless one flips the sign of the magnetic field, then the resulting skew-symmetry of the permeability tensor can also be absorbed in the now modified Onsager-Casimir reciprocity relationship with the added sign flip $\mathbf B \to -\mathbf B$ for both the static bias and the propagating field.
I think the answer already given is quite deep. I would only add that, at least for rank-2 tensors, with real-valued entries, symmetry makes them Hermitian, which ensures that their eigenvalues are real and that they are diagonalisible. So if I describe a material response with a symmetric rank-2 tensor, it means there exists a frame, that is a set of three orthogonal axis, such that material response to disturbance along these axis is a simple multiple of a disturbance magnitude, along the same axis.
