~あげる and ~あがる mean "finish."
But what are the differences in usage?
Should あげる strictly follow 他動詞？
For example, この本を読みあげた。
Should あがる strictly follow 自動詞？
For example, さっき走りあがった。
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After checking Compound Verb Lexicon, I can say the following.
上げる almost always follows another transitive verb, and the resulting verb is also transitive. One notable exception is こみ上げる, which is intransitive.
上がる follows many verbs regardless of its transitivity, but the resulting verb is intransitive.
Note that 上げる and 上がる cannot follow all verbs arbitrarily. They can follow only several dozens of verbs listed in the Lexicon above. You must say 走り終える or 走り終わる instead of 走り上げる or 走り上がる. (See also: The transitivity of (auxiliary) verbs)