In My car needs washing, washing is already passive. CGEL calls it a “concealed passive”:
2.3 Concealed passives (The house needs painting)
With a small number of catenative verbs, notably need, require,
deserve, and want, a gerund-participial may be passive while lacking the usual marking of the passive – we refer to this as the
concealed passive construction. Compare:
[ordinary passive]
[16] i a. The house needs to be painted.
b. These books want to be taken back to the library.
[concealed passive]
ii a. The
house needs painting.
b.
These books want taking back to the library.
Source: The Cambridge Grammar of the English Language