Two Bits solution
by Kegler

As described by 20 Across, DEPILATORY (and hinted at by the title, “Two Bits” being the response to “Shave and a Haircut”), six entries have a word meaning HAIR “cut” before entry.

Two Bits solution grid

Across
1P(ENT)AD (TEN anag)
5GROWTH (hidden)
9B(A + N + GAL)ORE
11A + OUD + AD (DUO rev)
13GLOCKENSPIEL (LEE PICKS LONG anag)
15T(RACING)S
16DEW + Y (WE'D rev)
17HAYES ("HAZE" hom)
18T(ERSE)R
20DEPILATORY
23PI(GS)TY
25A + SSAM (MASS rev)
27IDOL ("IDLE" hom)
28C(H + AIRMAN'S)HIP
31P(OUCH)ING
32SERENA (A SNEER anag)
33cENTER
34RESIST (SISTER anag)
35PLISS + E (SLIPS anag)
Down
1PAY THE PIPER (TYPE HAPPIER anag)
2NOVA (rev)
3TRICE + P.S.
4A + E + GI'S
5G(ON) + G
6R + US + SET
7WAITRESSES (SEE STRAW IS anag)
8HOLY ROMAN EMPIRE (MANY IMPLORE HERO anag)
10LIRA (rev hidden)
12DEWEY ("DO WE" hom)
14ENOL (rev)
19ROS(SIN)Ie
20DI + DOS (rev)
21dITCHES
22A LA + N
24GOULASHES (A HUGE LOSS anag)
25ANGEL (LANGE anag)
26mAIDEn
29MINT (2 defs)
30HA(T)S