Is Helvetica Neue, as sold by Adobe, the same thing as Neue Helvetica, as sold by Linotype?

Adobe lists the designer of Helvetica Neue as “Linotype Staff,” while Linotype appears to sell some Adobe fonts. The PostScript versions of both fonts appear to use the same “easy-to-use numbering system” in the font names.

Curiously, the Adobe OpenType version of Helvetica Neue doesn’t seem to have numbers in the font names, while the Linotype OpenType version does.

I’m writing a style guide, and I’m trying to figure out which of these fonts (if any) are interchangeable. Thanks for your help.

If you have any additional insights into differences between the two, or why this confusing situation exists, I’d love to know.

P.S. I originally posted this question on Ask Metafilter, and was advised to ask again here.