Intersections of Hyperconics in Projective Planes of Even Order

We show how to lift the even intersection equivalence relation from the hyperovals of PG(2, 4) to an equivalence relation amongst sets of hyperconics in π=PG(2, F). Here, F is any finite or infinite field of characteristic two that contains a subfield of order 4, but does not contain a subfield of order 8. Moreover, we are able to determine the number of points that two hyperconics in π will have in common provided some projective subplane of order 4 intersects both of them in hexads.