6

I've tried every variation I can think of, but the tag doesn't move.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
x=a
\raisetag{-10pt}
\end{equation}
\end{document}

demo

7

Quoting from the amsmath manual:

3.8 Adjusting tag placement
Placing equation numbers can be a rather complex problem in multiline displays. The environments of the amsmath package try hard to avoid overprinting an equation number on the equation contents, if necessary moving the number down or up to a separate line. Difficulties in accurately calculating the profile of an equation can occasionally result in number movement that doesn’t look right. There is a \raisetag command provided to adjust the vertical position of the current equation number, if it has been shifted away from its normal position. To move a particular number up by six points, write \raisetag{6pt}. This kind of adjustment is fine tuning like line breaks and page breaks, and should therefore be left undone until your document is nearly finalized, or you may end up redoing the fine tuning several times to keep up with changing document contents.

The command does nothing if the tag does not need to be shifted in the first place.

Example, where the box containing the formula is artificially made as wide as possible.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{gather}
\makebox[\displaywidth]{$\displaystyle x=a$}
\\
\makebox[\displaywidth]{$\displaystyle x=a$}
\raisetag{-10pt}
\\
\makebox[\displaywidth]{$\displaystyle x=a$}
\raisetag{20pt}
\end{gather}

\end{document}

enter image description here

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