A CliffordBasic calculation of the strain portion of the stress tensor, and an explicit demonstration that it is symmetric.

<< CliffordBasic`;
$SetSignature={3,0};

ClearAll[eta, epsilon, e1, e2, e3, h1, h2, h3, f, t1, t2, t3, t]
f = e1 e[1] + e2 e[2] + e3 e[3] + eta GeometricProduct[e[1,2,3],h1 e[1] + h2 e[2] + h3 e[3]] 

t1 = Grade[epsilon GeometricProduct[f, e[1], Turn[f]]/2,1];
t2 = Grade[ epsilon GeometricProduct[f, e[2], Turn[f]]/2,1];
t3 = Grade[ epsilon GeometricProduct[f, e[3], Turn[f]]/2,1];

t = {{ScalarProduct[t1,e[1]],ScalarProduct[t1,e[2]],ScalarProduct[t1,e[3]]},
{ScalarProduct[t2,e[1]],ScalarProduct[t2,e[2]],ScalarProduct[t2,e[3]]},
{ScalarProduct[t3,e[1]],ScalarProduct[t3,e[2]],ScalarProduct[t3,e[3]]}};
(*t // FullForm*)

format = {epsilon -> ϵ, eta -> η, e1 -> "(Subscript[E, 1])", e2 -> "(Subscript[E, 2])", e3 -> "(Subscript[E, 3])", h1 -> "(Subscript[H, 1])", h2 -> "(Subscript[H, 2])", h3 -> "(Subscript[H, 3])"};
t /. format // MatrixForm

Transpose[t] - t