I am not sure of the authenticity of the story that suggests that Lewis and Tolkien intended to write companion stories about space and time travel: Lewis' becoming the Space Trilogy (space travel) and Tolkien's the never-completed Lost Road (time travel). It seems like I got the idea from a respectable source, though I can't quite find it on second look.
It seems, however, like Tolkien's and Lewis's stories converge more than one would think on first glance. For the first, "Numenore" is referenced numerous times in
That Hideous Strength as a place from which wisdom and knowlege come. The connection, which Tolkien intentionally drew, between Numenor and Atlantis is barely referenced at all, Lewis skips directly over it and references Numenor itself. He connects Numenor with the knowledge of Merlin, specifically, and I believe there are hints of Tol Eressea there as well, in the links between the Arthur legends, Avalon, Avallone, and Eressea (another parallel drawn intentionaly by Tolkien and borrowed by Lewis).
For the second, the prophecy of the coming of Eru as an incarnate to Middle-Earth in the Athrabeth seems to echo in the story of Maleldil's coming to Thulcandra told in
Perelandra. Both are clearly Christ-pictures, almost allegorical (though I hate to use that word in reference to Tolkien, knowing what fires it can start

). Lewis's is certainly just a "different language" reference to Christ, since his stories are set in "this world, this time" just farther out in the solar system. Tolkien's is a little more difficult to place, as Middle Earth is supposed to be this world, but in a far more legendary sense. For instance, in ME there is no planet Venus--the morning star is Earendil. Makes it thorny to say that the Lost Road and
Perelandra are set in the same world, since Perelandra is set on Venus.
I also suspect that the indwelling
eldil of the various planets are comparable, and perhaps meant to be compared to, the Valar and Maiar occupying Tolkien's world. The description of their physical forms is interesting, the Eldil cannot be seen in this world, but can be seen by those who know how to look for them as flickers of white or gold light, or sometimes as pillars of light. The Valar when unclothed walk as white flames and are visible on the "other side" as radiant beings.
The question, and maybe one more for research than discussion: did Tolkien know about and endorse or even reciprocate these back and forth references? Or did Lewis act more or less on his own? Or am I simply overanalyzing the links between the two works?
Sophia