In unicode-math-table.tex the math class of many symbols is defined by lines like:
\UnicodeMathSymbol{"026AC}{\mdsmwhtcircle}{\mathord}{medium small white circle}%
Now if I want to use this particular circle as binary operator for function composition I have to change its math class to \mathbin. What I would like to do in the preamble is something like:
\usepackage{unicode-math}%
\UnicodeMathSymbol{"026AC}{\mdsmwhtcircle}{\mathbin}{medium small white circle}%`
Thereby overwriting unicode-math’s defaults. However the marco \UnicodeMathSymbol seems only to be available internally, when unicode-math processes unicode-math-table.tex.
Of course I can copy unicode-math-table.tex to a local texmf-tree, modify it and achieve the desired result. But I would prefer to keep this change local to the document and make the change in the preamble.
How can I overwrite unicode-math’s defaults with regard to the symbols math class?

unicode-mathhave changed the internal command used in the second solution. it should now be\__um_process_symbol_noparse:nnn(the command is prefixed with a double underscore. – ArTourter Aug 03 '15 at 15:37:-(There should be some interface to associate a Unicode point to a command with the math type. – egreg Aug 03 '15 at 16:54