Also, the blog post in the submission omits a major detail: the on-orbit docking maneuvers for the CSM to mate with the LM. A minor detail is that the Saturn V's third stage performed TLI (trans-lunar injection) and it actually impacted the Moon. After this TLI, the LM and the CSM were flying free in space, with a bit of separation, and it was the CSM pilot who needed to turn 180° and nose-in to the LM in order to be in the proper configuration for the hypothetical Moon landing.

It was an unusual configuration for Apollo 13, to say the least, because of course they did not land on the Moon, but also because the "base/legs" part of the LM wouldn't be "left behind" on the lunar surface, so they sort of lugged it around awhile. I don't know the exact sequence of jettisoning that base, but they certainly relied on the LM "head" as a lifeboat and a source of additional life-support functions.