Both 上がる and 挙がる are read the exact same way — as あがる — and in the
dictionaries I looked them up, they appear under the same entry, and seem to
show up as alternate spellings of the same word.

Is is the case that either spelling works on all occasions (meaning they're
completely interchangeable), or are the spellings used in different
contexts/circumstances?

[This question](https://japanese.stackexchange.com/a/12118/1330) does mention
some variations, but since it is a rather broad one, wanted to get some more
clarity around the specific nuance between the two writings.
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question
nuances spelling homophonic-kanji
What is the difference between 上がる and 挙がる?
Different characters are used for different meanings or contexts.

スーパー大辞林 has this note on its (only) entry for あがる

> あがる《上・揚・挙・騰》  
>  「上がる」は“上の方へ移動する。高くなる”の意。「屋上に上がる」「気温が上がる」「雨が上がる」  
>  「揚がる」は“揚げ物ができる。かかげられる。高まる”の意。「天ぷらが揚がる」「日の丸が揚がる」「歓声が揚がる」 「意気が揚がる」  
>  「挙がる」は“示される。犯人がつかまる。上へ動く”の意。「証拠が挙がる」「犯人が挙がる」「手が挙がる」  
>  「騰がる」は“物価が高くなる”の意。「上がる」とも書くが，仮名書きも多い。「物価が騰がる」

* * *

上げる is the most basic of all and is used for the general sense of raising
something or moving something up. 挙げる has the implication of raising something
or bringing something up to indicate something or so something is noticed.
例を挙げる is arguably the most common usage. Even 手を挙げる is understood slightly
differently from 手を上げる. The former seems to suggest you raise your hand for
some purpose, for example as a sign that you have something to say, whereas
the latter simply refers to the physical motion.
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answer