  Ba\~nuelos and Bogdan (2004) and Bogdan, Palmowski and Wang (2016) analyse
the asymptotic tail distribution of the first time a stable (L\'evy) process in
dimension $d\geq 2$ exists a cone. We use these results to develop the notion
of a stable process conditioned to remain in a cone as well as the the notion
of a stable process conditioned to absorb continuously at the apex of a cone
(without leaving the cone). As self-similar Markov processes we examine some of
their fundamental properties through the lens of its Lamperti-Kiu
decomposition. In particular we are interested to understand the underlying
structure of the Markov additive process that drives such processes. As a
consequence of our interrogation of the underlying MAP, we are able to provide
an answer by example to the open question: If the modulator of a MAP has a
stationary distribution, under what conditions does its ascending ladder MAP
have a stationary distribution?
  We show how the two forms of conditioning are dual to one another. Moreover,
we construct the recurrent extension of the stable process killed on exiting a
cone, showing that it again remains in the class of self-similar Markov
processes.
  In the spirit of several very recent works, the results presented here show
that many previously unknown results of stable processes, which have long since
been understood for Brownian motion, or are easily proved for Brownian motion,
become accessible by appealing to the notion of the stable process as a
self-similar Markov process, in addition to its special status as a L\'evy
processes with a semi-tractable potential analysis.
