> Does the C89 standard allow this code?
> 
> int q = 1;
> int p = (unsigned)q;
> I've never seen an unsigned cast without a type.

Yes, that's fine; it's a different spelling of "unsigned int".
In C99, 6.7.2p1 defines the following groups as equivalent:
- short, signed short, short int, or signed short int
- unsigned short, or unsigned short int
- int, signed, or signed int
- unsigned, or unsigned int
- long, signed long, long int, or signed long int
- unsigned long, or unsigned long int
- long long, signed long long, long long int, or
            signed long long int
- unsigned long long, or unsigned long long int

Specifiers may occur in any order, so you may also write
"int short unsigned".