Most of what I'm taking here is from theoi.com which is an excellent myth resource
The god of prophecy. Apollo exercised this power in his numerous oracles, and especially in that of Delphi. (Dict. of Ant. s. v. Oraculum) The source of all his prophetic powers was Zeus himself (Apollodorus states, that Apollo received the mantikê from Pan), and Apollo is accordingly called "the prophet of his father Zeus." (Aeschyl. Eum. 19).
Zeus was not swallowed by Cronos but exchanged for a stone put in swaddling clothes which was later spit out to become the stone at the oracle at Delphi. Apollo later defeated a serpent that guarded Delphi and took control of it.
The swaddling of the stone is a clear reference to the skin of the child and in psychoanalysis we know of skin erotism and what would be important in ego development (ego in the sense of the subject in Lacanian terms) as the differentiation of inside and outside.
This connection to the skin is also seen in the myth of Marysyas:
MARSYAS was a Phrygian Satyr who first composed tunes for the flute. He obtained his instrument from Athena, who had invented the device but discarded it in her displeasure over the bloating effect on the cheeks. Later, in hubristic pride over the new-found music, Marsyas dared challenge the god Apollon to a contest. The Satyr inevitably lost, when, in the second round, the god demanded they play their instruments upsidedown--a feat ill-suited to the flute. As punishment for his presumption, Apollon had Marsyas tied to a tree and flayed him alive. The rustic gods in their pity then transformed him into a mountain stream.
In addition there is a connection of Apollo to Helius:
These characteristics of Apollo necessarily appear in a peculiar light, if we adopt the view which was almost universal among the later poets, mythographers, and philesophers, and according to which Apollo was identical with Helios, or the Sun. In Homer and for some centuries after his time Apollo and Helios are perfectiy distinet. The question which here presents itself, is, whether the idea of the identity of the two divinities was the original and primitive one, and was only revival in later times, or whether it was the result of later speculations and of foreign, chiefly Egyptian, influence.
Helius seems to be a forerunner of Zeus as the ruler of the sky, but as a titan he is still earlier than Zeus. This suggests that Zeus comes into his own in the titanomachy and the battles against Typhon which places his importance at the "anal" stage:
Typhon hamstrung, sank to his knees in despair… for miles the whole great earth was enkindled by the blast of heavenly wind
From the waist down he was a tangle of hissing vipers, he had majestic wings, and shaggy hair covered his body
The references to the heavenly wind and the lower body being vipers point to the muscular tensions involved in the lower body in the anal stage. Typhon being buried under a mountain point to obssesionalism and the schizoid traits manifested in Hephaestus. Typhon is thus the chthonic figuration of volcanic forces, as Hephaestus (Roman Vulcan) is their "civilized" Olympian manifestation. Dwarves are also represented as dwelling in mountains and caring only for the treasures they mine there in a similar parallel.
If Hephaestus is the Olympian version of the schizoid and the epistemophillic drive of binaries (up down, left right, etc.) then Helios betrays a pre-verbal, pre-walking, epistemological relation to the world.
Apollo's gift of prophecy seems to relate to an instinctual renunciation in the period before he boundaries of the skin are in place. The failure of the drive or ideal would mean that the skin boundary comes into prominence where otherwise "the gaze" of Apollo could somehow have a connection with others in an early energetic sense. I hesitate to talk of prophecy being a real ability but I respect the idea as a metaphor for a pre-verbal epistemological power. The connection between twins in which one knows across great distances that something has befallen the other is something that I haven't ruled out. There seems to be an interesting triangulation in that people are paired across distances by the triangulation of the sun (of Helius). At the least there is certainly the necessity that for music to communicate feeling/moods the boundaries between inside and outside must be broken.