I have learned that chirality is a concept, that appears for $(A,B)$ representations of the Lorentz group, where $A\neq B$. 

An example would be a Dirac spinor, corresponding to the representation $(\tfrac{1}{2},0)\oplus(0,\tfrac{1}{2})$, where we can identify left- and right-chiral components. 

[Wikipedia][1] lists the electromagnetic field strength tensor $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ as transforming under the $(1,0)\oplus(0,1)$ representation of the Lorentz group. 

> Supposing my first sentence is true, where can I see chirality in the electromagnetic field strength tensor? 

  [1]: https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group#Common_representations