  Motivated by the significantly improved scalability of optically-trapped
neutral-atom systems, extensive efforts have been devoted in recent years to
quantum-state engineering in Rydberg-atom ensembles. Here we investigate the
problem of engineering generalized (``twisted'') $W$ states, as well as
Greenberger-Horne-Zeilinger (GHZ) states, in the strongly-interacting regime of
a neutral-atom system. We assume that each atom in the envisioned system
initially resides in its ground state and is subject to several external laser
pulses that are close to being resonant with the same internal atomic
transition. In particular, in the special case of a three-atom system
(Rydberg-atom trimer) we determine configurations of field alignments and
atomic positions that enable the realization of chiral $W$ states -- a special
type of twisted three-qubit $W$ states of interest for implementing
noiseless-subsystem qubit encoding. Using chiral W states as an example we also
address the problem of deterministically converting twisted $W$ states into
their GHZ counterparts in the same three-atom system, thus significantly
generalizing recent works that involve only ordinary $W$ states. We show that
starting from twisted -- rather than ordinary -- $W$ states is equivalent to
renormalizing downwards the relevant Rabi frequencies. While this leads to
somewhat longer state-conversion times, we also demonstrate that those times
are at least two orders of magnitude shorter than typical lifetimes of relevant
Rydberg states.
