The order of the Siddham letters is not an invention of the script and is a great deal older than it. It comes from the Indian grammatical tradition, whose analysis of the sounds of Sanskrit by point and manner of articulation was already complete and highly refined before any of the surviving Brahmic scripts took the shape they now have. Panini, whose grammar is the central work of that tradition, describes a language rather than a way of writing it, and the phonetic treatises standing beside the grammar set out the classification the letter table simply follows.
That is why the arrangement is unlike an alphabet. The Mediterranean order is inherited and arbitrary, a sequence of letters whose original rationale, if there was one, is lost. The Indian order is a claim: these are the points at which the mouth closes, these are the features that cross them, and here is every combination the two produce. A student reciting it is reciting a theory.
The theory travelled better than the script did. Chinese and Japanese scholars studied the table as an analysis of speech and used it on their own languages, and the standard ordering of the Japanese kana is the visible result, a grid by consonant and vowel with the consonants running in the Sanskrit sequence. Indian phonetics reached Japan as a working tool, carried there by monks who wanted to pronounce a mantra correctly, and stayed as the frame for describing a language it was never designed for.