  We introduce a novel definition of orientation on the triples of a family of
pairwise intersecting planar convex sets and study its properties. In
particular, we compare it to other systems of orientations on triples that
satisfy a so-called interiority condition:
$\circlearrowleft(ABD)=~\circlearrowleft(BCD)=~\circlearrowleft(CAD)=1$ imply
$\circlearrowleft(ABC)=1$ for any $A,B,C,D$. We call such an orientation a P3O
(partial 3-order), a natural generalization of a poset, that has several
interesting special cases. For example, the order type of a planar point set
(that can have collinear triples) is a P3O; we denote a P3O realizable by
points as p-P3O.
  If we do not allow $\circlearrowleft(ABC)=0$, we obtain a T3O (total
3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O
realizable by points, a p-T3O, is the order type of a point set in general
position. Despite these similarities to order types, P3O and p-T3O that can
arise from the orientation of pairwise intersecting convex sets, denoted by
C-P3O and C-T3O, turn out to be quite different from order types: there is no
containment relation among the family of all C-P3O's and the family of all
p-P3O's, or among the families of C-T3O's and p-T3O's.
  Finally, we study properties of these orientations if we also require that
the family of underlying convex sets satisfies the (4,3) property.
