Is linear-time reduction symmetric?
No, it is not symmetric.

Here's my counter-example: Consider the language $L_1 = \{A:A'| A'$ is $A$  sorted$\}$
and $L_2 = \{A:a| a$ is the largest element in $A\}$.

We can solve the max-element problem with a constant number of calls to a machine solving sorting. Just sort $A$ then take the first element. But we can't solve sorting with a constant number of calls to a machine solving max-element (otherwise, we could sort in $O(n)$, which is known to be impossible).