S. E. Martin, A Reference Grammar of Japanese, Yale University Press, 1975, pp. 358-9.
Desiderative verbals are made from the (expanded) desiderative sentence by attaching -garu, using a suffix -garu- that derives verbs from certain adjectives and adjectival nouns (§7.3) but here is attachable to ANY desiderative from ANY underlying verbal except Aru; we find Ari-tai (in written Japanese, at least) but not *Ari-ta-garu. The accent pattern of -ta-garu is optionally (1) always a tonic, or (2) tonic only when the underlying verb is tonic; in this book we choose the latter option, since it is maximally differentiating. (The accent pattern of the desiderative adjectives V-i-tai is similar, having the option of being always tonic or being tonic only when the underlying verb is tonic, and we have chosen the latter option in this book.)
The verbal desideratives have a strong meaning 'desires, is eager to' and in sentences of straightforward reference they are seldom -- if ever -- used of one's own desires: Tookyoo ni uturi-tai kimoti wa atta si, uti no oku-san mo uturi-ta-gatte ita n desu 'I was in the mood to move to Tokyo and my missus wanted to move, too' (SA 2671.30c -- notice the unexpected use of uti no oku-san for kanai 'my wife'); Sigoto no owari-goro, kano-zyo wa watasi no moderu o yame-ta-gatte ita 'At the end of the job, she wanted to quit as my model' (SA 2658.133b). but in sentences that imply a shift of viewpoint -- so that from YOUR point of view, say, "I" temporarily become "you" -- the first person can take the verbalization: Watasi ga sonna koto o si-ta-garu to omoimasu ka 'Do you think that I want to do such a thing?'; Watasi ga iki-ta-gatta mon da kara turete itte kuremasita 'Seeing how badly I wanted to go, he took me with him'.
The sentence often appears in the continuous aspect -ta-gatte iru, §9.2.4.(1); the desiderative verbalization would seem to be used as a durative-stative verb like sugureru 'excels' (§3.12), but see p. 360 (§7.3) for a more precise characterization. It is possible to add the structure -ta-gatte iru to a sentence that has already undergone an auxiliary conversion to V-te iru, and then it is unclear whether the "desire" has reference to the entire phrase, to the auxiliary, or to the underlying verb: Asa kara ban made sake o nonde i-ta-gatte iru 'he keeps wanting to drink (or: He wants to keep drinking) liquor from morning till night'.
Notice the chain VERBAL (as INFINITIVE §9.1) → ADJECTIVE (-ta-i) → VERBAL (-ta-gar-u). The new verbal retains some of the grammatical properties of the immediately underlying desiderative adjectival: it can not be turned into a command, a request, or an exhortation, for desires are involuntary, and not subject to human control.
The object of the underlying verb will always retain its o-marking. Compare Ano hon o yomi-ta-gatte iru 'He is eager to read that book' with Ano hon o/ga yomi-tai n desu 'I want to read that book'. The transitivity or intransitivity of the underlying verb is not affected by the verbalization of the desiderative: (uni e) kaeri-ta-garu 'is eager to go (home)' remains as intransitive as kaeru 'goes (home)'.
Examples (from KKK 3.259-60): ... Ano musume ga, omote kara hairi-ga-garanakatta kara desu 'It is because tha girl did not want to enter from the front way'; ... yatara ni hue o huki-ta-garu '... they =certain referees] are overly eager to blow their whistles'; Hakkiri mono no mieta hito da to omoimasu ga, hizyoo ni manabi-ga-gari, siri-ta-gatta, soo iu imi de tankyuu-sin no atta hito da to omoimasu 'He is a man of clear discernment, I believe, and -- extremely desirous of learning and desirous of knowing -- he is in that sense a person possessed of the spirit of inquiry, I believe'; Motiron Sakamaki no yoo na, gyosyok-ka de tootte iru otoko ga mi-ga-garu sibai nara, ooyoso kentoo wa tuku 'Of course if it's play thaat a man liek Sakamaki, known as a lecher, is eager to see, I can pretty well guess [the content]'; Sonna ni tabe-ta-gareba, donna hahaoya d'atte tabesasezu ni wa irarenai to omoimasu 'When they are so eager to eat, it seems to me that hardly any mother can resist feeding them'.
The V-i-ta-garu formation originated in Kamakura times; through the Muromachi period the negative version was usually made by applying the verbalization to the negative auxliary: V-i-tai → V-i-ta[k]ku nai → V\-i-ta'u na-garu. The modern negativization (V-i-tai → V-i-tagaru → V-i-tagaranai or -tagaranu) dates from the Edo period. Cf. Y 304.