In Japan the study of Siddham became a field with a name, shittan-gaku, and a literature of its own. It was not a language course. Very few of its students learned to read Sanskrit as a language, and that was not what it was for. What was taught was the letters, their order, their sound values, the way they combine into conjuncts, the way the vowel marks attach, and the reading traditions by which each syllable was pronounced in a Japanese mouth.
That may look like an odd half measure, and it follows exactly from what the script was needed for. Mantras and dharani were transmitted as sound and written as sound, so a practitioner needed to produce the syllables correctly and to recognise them on a page, on an image or on a stone. Grammar and vocabulary were a separate matter and were largely left to the translations, of which the Chinese canon held an enormous quantity.
The discipline generated manuals, tables, commentaries and dictionaries, and it kept going for centuries after the Indian tradition it drew on had changed beyond recognition. It also left marks outside itself. The ordering of the Japanese kana follows the Sanskrit arrangement of sounds by place and manner of articulation rather than any native principle, which is the clearest evidence that the analysis behind this alphabet was studied in Japan as a description of speech and not merely copied as a set of shapes.