The pair 預ける・預かる is actually not irregular at all: it belongs to a class of verbs of conveying (mostly giving and saying) that behaves differently in this regard in many languages.
To see that we have to look at the relations between transitive and intransitive verbs (as well as passive and active verbs) from a linguistic point of view - this is something that linguists call Valency (or Valence). From this point of view, verbs have a set of arguments that specify all of the participants that took part in the event described by the verb. Each of the arguments is marked differently than the other (in English using word order and prepositions, in Japanese with particles alone).
The verb 預ける has three arguments (not all of them have to be explicitly specified of course):
- {giver}[が]{ }{recipient}[に]{ }{deposit}を預ける
The verb 預かる has three arguments specifying the very same roles, but they are marked differently:
- {recipient}[が]{ }{giver}[に]{ }{deposit}を預かる
Normally, pairs of transitive and intransitive verbs also have the marks on their arguments switched, but instead of switching 'the subject' (marked by が) with the argument marked by に, they switch it with the argument marked by を, which is usually (and rather inaccurately) called 'direct object', due to the bad influence of traditional Latin grammar.
- {discoverer}[が]{ }{discovered}を見つける
- {discovered}[が]{ }{discoverer}に見つかる
Verbs of saying and giving, however, work in a different ways. I can't think of -eru/aru pairs of such verbs, so I'll give examples of active/passive pairs instead:
The verb argument pattern of the verb 与える:
- {giver}[が]{ }{gift}[を]{ }{recipient}に与える
has at least two possible pair patterns with the passive verb 与えられる:
- {recipient}[に]{ }{giver}[から]{ }{gift}が与えられる
- {recipient}[が]{ }{giver}[[に・から]{ }]{gift}を与えられる
The second option is the which interests us, because almost the same thing happens with 預かる:
(私が誰かに)子猫のチーを預けました。 I entrusted Chii the kitten to someone.
(私が誰かに)子猫のチーを与えた。 I gave Chii the kitten to someone.
(私が誰かに)子猫のチーを預かりました。 Someone entrusted me to take care of Chii the kitten.
(私が誰かに)子猫のチーを与えられた。 Someone gave me Chii the kitten.