Were 'th' and 'ch' aspirated in classical Latin?

Question

I have been taught that 'th' and 'ch' were pronounced just like 't' and 'c' in classical Latin, with no aspiration. The answer to this earlier question confirms that 't' and 'c' had indeed little or no aspiration. But did 'th' and 'ch' really lack aspiration as well? Was there any distinction between 't' and 'th' or 'c' and 'ch'?

The combinations 'th' and 'ch' mostly appear in Greek loan words, but for example pulcher is — as far as I know — not a Greek loan. The corresponding Greek letters θ and χ were aspirated, and I have understood that aspiration was a way of showing of high education, not a widespread feature in Latin pronunciation.

In some rare cases aspiration is called for, like in Catullus' poem to Harrius. There one needs strong aspiration to make a difference between commoda and chommoda. Rare examples like this do not necessarily imply that every occurrence of 'ch' and 'th' is aspirated.

Answer 1

W. Sydney Allen, not unexpectedly, has the answer in Vox Latina, 26–27:

The digraphs ph, th, ch represented aspirated voiceless plosives—not unlike the initial sounds of pot, top, cot respectively.

They were indeed aspirated, and this is due to Greek influence. They were not found in the oldest inscriptions (p, t, and c sufficed), and initially, beginning in the 2nd century BC, appeared only in Greek loan words:

They are then used, and become standard, primarily in transcribing Greek names and loan-words containing aspirated plosives (φ, θ, χ), e.g., Philippus, philtrum, Corinthus, cithara, thesaurus, Achaea, bacchanal, machina, chorus; and in such cases it is likely that educated Roman speakers in fact reproduced the Greek aspirates with more or less fidelity.

But it wasn't long until native Latin words, like pulcher, began using them. Other examples include lachrima, sepulchrum, and triumphus.

Cicero discusses an actual pronunciation change taking place during his time in Orat. 48.160; he accepts the new pronunciation of pulcher but not of sepulchrum. Allen argues that the shift was not merely "fashionable misapplication of Greek speech-habits," but that it was a "natural environmental development in Latin itself" that began using the introduced diagraphs to mark aspiration.

Allen concludes that ph, th, and ch were certainly more aspirated than p, t, and c, and reminds readers that they were not fricatives (i.e., photo, thick, loch) in classical Latin.