I came across an interesting passage in the pseudo-Aristotelian treatise, Mechanics. The author asks:

§ 12 Why is it that a missile travels further from a sling than from the hand although he who casts it has more control over the missile in his hand than when he holds the weight suspended? Further, in the latter case he moves two weights, that of the sling and the missile, while in the former case he moves only the missile.
Is it because he who casts the missile does so when it is already in motion in the sling (for he swings it round many times before he lets it go), whereas when cast from the hand it starts from a state of rest? Now any object is easier to move when it is already in motion than when it is at rest. Or, while this is one reason, is there a further reason, namely, that in using a sling the hand becomes the centre and the sling the radius, and the longer the radius is the more quickly it moves, and so a cast from the hand is short as compared with a cast from a sling? (852a39-b10)

This passage is less interesting for what it tells us about the science of slinging than what it tells us about the technique that ancient Greek slingers used. The passage claims that the slinger swings it around many times before letting it go. This suggests that something like the helicopter or Balaeric style might have been the most common style used by Greek slingers.

I agree that this is the implication of the passage. I gathered six  references from ancient literary sources in this thread, with some notes on critical method:

http://slinging.org/forum/YaBB.pl?num=1407020188

(page 4 also gathers some visual evidence)

and I repeat them here. Most of them are, in fact, from Latin literature, so you may want to decide how relevant they are. The internet search engines for all of Greek literature give you about 500 occurrences of σφενδονή, sling, and cognates; of course, you have to add paraphrases, etc.


Literary evidence
There are only two I know of. They are later than the period we're talking about.

1. Virgil, Aen. 9.558.
stridentem fundam positis Mezentius hastis
ipse ter adducta circum caput egit habena
et media aduersi liquefacto tempora plumbo
diffidit
Mezentius lays down his spears, and thrice around his head drives the buzzing sling
with taut thong, and with molten lead splits the forehead of his foe,
It describes 3 rotations above the head, slinging lead. Problem: it's in an epic poem, which describes lots of things, some quite exaggerated. . But precision and realism might be aimed for here, in contrast with the more outré fighting scenes


2.Vegetius 2.23
Adsuescendum est etiam, ut semel tantum funda circa caput rotetur, cum ex ea emittitur saxum.

The habit should be given, to spin the sling only once around the head, when it is used to throw a stone.

The same adverb, "around", being used in both passages, helo. or abbreviated helo. is being described here, since "three times" must describe multiple rotations, with a low rotor. This can be considered absolutely certain, on linguistic grounds.

and

3
Re: How did the ancient sling ?
Reply #81 - Sep 7th, 2014 at 7:40pm   In the treatise on Mechanics attributed to Aristotle (but certainly not by him), the question is raised-- why do projectiles fly further from the sling than when thrown by hand ?

One of the explanations is
ὅτι ἐν μὲν τῇ σφενδόνῃ κινούμενον τὸ
βέλος ῥίπτει ὁ βάλλων (περιαγαγὼν γὰρ κύκλῳ πολλά-
κις ἀφίησιν), ἐκ δὲ τῆς χειρὸς ἀπὸ τῆς ἠρεμίας ἡ ἀρχή·

(852A)

"because the thrower throws with the sling a projectile in motion (for he looses it having led it round in a circle many times), but by hand from an immobile starting point"

Perhaps Archytas of Tarentum, late C5th BCE (Thomas Winter); perhaps much later, e.g. C3rd BCE. Anyway, the earliest description of the movement of a sling, by a natural philosopher.

http://en.wikipedia.org/wiki/Mechanics_(Aristotle)

4. From the lexicon of Pollux (Roman era work on Classical Greek usage). 1.175

ἐρεῖς δὲ τὰς σφενδόνας πληρώσασθαι, τὰς σφενδόνας στρέψαι, (5)
τὰς σφενδόνας ἐναγκυλίσασθαι

You can also say  "fill your slings", "twist / spin/ turn your slings", and "fit your slings to their thongs / 'thong' your slings'.

No idea what the third one means. It reads like military orders.

5.Another reference, Silius Italicus 1.314ff

hic crebram fundit Baliari uerbere glandem
terque leui ducta circum caput altus habena            315
permissum uentis a<b>scondit in aera telum,

One hurled volleys of bullets with Balearic sling: standing erect, he brandished the light thong thrice round his head, and launched his missile in the air, for the winds to carry

It's a poet, writing about Hannibal in the first century CE. Like Virgil, he believes that three rotations around the head is the way to go

6.  Vegetius, 3.14
Funditores sunt qui fundis lino uel saetis factis - has enim dicunt esse meliores - contorto circa caput brachio dirigunt saxa.

Slingers are those who throw stones with slings made of linen or coarse hair (the latter are said to be the best) by turning / twisting their arm around the head.

(For me, this, again, is about revolutions around the head, i.e helo style).

[edited to add in the actual figure for occurences of slinging in Greek literature-- I had to check]