I've just noted that [tja] and [tʲa] sound exactly the same. It seems that [ʲ] would only be relevant in classrooms when having isolated pronunciations such as [tʲ], [kʲ], [dʲ] due to the fact we don't take [j] to be syllabic.
Actually, [tja] doesn't even sound exactly the same across languages. There are theoretically three ways to notate a "palatalized t" in IPA: [tja, tʲa, t͡ja]. The options [tʲa, t͡ja] commit one to the claim that this is a single segment, and [tja] does not have that commitment. There are various reasons to want to be able to indicate that a supposed bit of sound is a single segment versus a set of segments, the most powerful one being contrast (if a language contrasts the sequence [tj] and the single segment, which could be written as either [tʲa, t͡ja], though usually people write [tʲ]. The main idea is that in a sequence [tj], the sequence of gestures defining [t] vs [j] would overlap a bit but they would not be simultaneous, whereas in [tʲ] they would be simultaneous. However, simultaneity is hard to document.
In the case of labiovelars in Lushootseed, there are phonemes [kʷ, gʷ, xʷ, qʷ, χʷ] which seem to satisfy criteria for phonetic and phonological unity. They appear in the coda and the preceding vowel is strongly coarticulated with the following lip rounding, for example in šəgʷɬ "path", the lip-rounding of gʷ is quite audible in the middle of ə. Analyzing these as clusters would make the rules of syllable structure more complex, since glides are generally not found between consonant.
It is typically difficult to come up with good arguments for the treatment of postconsonantal glides. Distributional arguments generally support a sequence analysis of [tw] in English, whereas in Mandarin, I can't say that I've seen a compelling argument to support the [kw] vs [kʷ] analysis.