  Twin groups are planar analogues of Artin braid groups and play a crucial
role in the Alexander-Markov correspondence for the isotopy classes of immersed
circles on the 2-sphere without triple and higher intersections. These groups
admit diagrammatic representations, leading to maps obtained by the addition
and deletion of strands. This paper explores Brunnian twin groups, which are
subgroups of twin groups composed of twins that become trivial when any of
their strands are deleted. We establish that Brunnian twin groups consisting of
more than two strands are free groups. Furthermore, we provide a necessary and
sufficient condition for a Brunnian doodle on the 2-sphere to be the closure of
a Brunnian twin. Additionally, we delve into two generalizations of Brunnian
twins, namely, $k$-decomposable twins and Cohen twins, and prove some
structural results about these groups. We also investigate a simplicial
structure on pure twin groups that admits a simplicial homomorphism from
Milnor's construction of the simplicial 2-sphere. This gives a possibility to
provide a combinatorial description of homotopy groups of the 2-sphere in terms
of pure twins.
