ROLE OF REAGENT.........

inversion in the product certainly ring the bells 4 the SN₂ and SN₁ is ruled out
i think that
SOCl₂ + Pyridine and HBr or HI should bring the inversion of the products as per the reason mentioned [1]

R--OH + PCl₅ , PBr₃ , SOCl₂ or SO₂Cl₂ -----with base-------> X--R

R--OH + PCl₅ , PBr₃ , SOCl₂ or SO₂Cl₂ -----without base-------> R--X

eg. of base : Pyridine , R₃N , etc.

And as mani said , Inversion means S_N² (BUT with few exceptions as above.)

S_N¹ means Retention in configuration along with Racemization.

arey yaar SOCl₂ aur pyridine ke combination main bhi inversion of configuration ke chances hote hain... SN_i mechanism....

I think in SNⁱ mechanism there is only retention.

are yaar when we use socl₂ with pyridene there will be generation of strong nucleophile cl^- with the reaction of formed hcl from h⁺,cl^- in the first step and pyridene hence the reaction changes to sn² and there will be inverted pdct