What binary relation is neither symmetric, nor asymmetric nor antisymmetric?

For a relation $R$ to be symmetric, every ordered pair $(a,b)$ in $R$ will also have $(b,a)\in R$.

For a relation to be asymmetric, every ordered pair $(a,b)\in R$ does not have $(b,a)\in R$.

For a relation to be antisymmetric, if both $(a,b)$ and $(b,a)$ are in $R$ then $a=b$.

So we want $R$ such that for some $a\neq b$, $(a,b)$ and $(b,a)$ are both in $R$ this makes sure $R$ is neither asymmetric nor antisymmetric; but at the same time we want some $(c,d)\in R$ such that $(d,c)\notin R$, as this will ensure that $R$ is not symmetric.

In either case, we need witnesses in $R$ to prove that it is not symmetric or asymmetric. Therefore it cannot be empty.

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I will leave the grueling details of writing down such $R$ for you.

$\varnothing$ is the (unique) example of a relation that is symmetric, asymmetric and antisymmetric (so the opposite of what you are looking for).

E.g. a relation $R$ is not symmetric if an ordered pair $\langle a,b\rangle$ exist with $\langle a,b\rangle\in R$ and $\langle b,a\rangle\notin R$.

This is evidently not true for $R=\varnothing$ (it contains no ordered pairs) so we conclude that $\varnothing$ is symmetric.

Likewise it can be shown that $\varnothing$ is asymmetric and antisymmetric.

The implications involved are vacuously true.

In the empty relation, it is impossible for $xRy$, so that whenever $xRy$ it is true that $yRx$ - you have a statement about elements of the empty set, and there aren't any such elements to make the statement false.