Shavuot and Shemini Atseret - Two of a Kind?

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What do Shavuot and Shemini Atseret have in common? A brief consideration of these holidays reveals that they bear an unmistakable resemblance to one another on a variety of levels:

1) Both are observed for a single day only (two days in the Diaspora).

2) Both are "appended", as it were, to seven day holidays - Shavuot is the culmination of the Omer count begun on Pesah, and Shemini Atseret represents the culmination of Sukkot.

3) Both are referred to as "Atseret" - Shemini Atseret is designated as such by the Torah itself (hence its name), whereas, in Rabbinic parlance, "Atseret" is the term used for Shavuot.

4) Both holidays lack any concrete ritual expression or seasonal commandment (outside of the special offerings brought in the Temple, that is). Unlike Passover, which is associated with the consumption of matsa, and Sukkot, which revolves around dwelling in the Sukkah and waving the Four Species, Shavuot and Shemini Atseret do not obligate us in any positive mitsvot at all.

5) Both Shavuot and Shemini Atseret are thematically linked to the study of Torah. Shavuot commemorates the Revelation at Sinai, while Shemini Atseret is "Simhat Torah" - the day that we celebrate our completion of yet another annual cycle of Torah readings.

What is the reason for these remarkable similarities?

Stay tuned for a fascinating answer based upon a cryptic commentary from Nachmanides.

Meanwhile, of course, educated guesses are welcome in the comments to this post.