Abstracting a Password Service
This looks quite good, but as usual improvements can be made.  


  private readonly PasswordHasher hasher;  


instead of using the implementation, you should use the interface  

private readonly IPasswordHasher hasher;  


I don't see why you want to use both IPasswordHasher and IPasswordService. If your application is built upon IPasswordService you can just skip the implementation of the IPasswordHasher interface like    

public class PasswordServiceAdaptor : IPasswordService
{
    private readonly IPasswordHasher hasher;

    public PasswordServiceAdaptor()
    {
        this.hasher = new PasswordHasher();
    }

    string IPasswordService.HashPassword(string password)
    {
        return this.hasher.HashPassword(password);
    }

    bool IPasswordService.VerifyPassword(string hashedPassword, string testPassword)
    {
        var result = this.hasher.VerifyHashedPassword(hashedPassword, providedPassword);

        return result == PasswordVerificationResult.Success;
    }
}


As it seems from the comments that you need to implement both interfaces, you should consider to just call the hasher directly without the call to AsPasswordHasher().  

public class PasswordServiceAdaptor : IPasswordService, IPasswordHasher
{
    private readonly IPasswordHasher hasher;

    public PasswordServiceAdaptor()
    {
        this.hasher = new PasswordHasher();
    }

    string IPasswordHasher.HashPassword(string password)
    {
        return this.hasher.HashPassword(password);
    }

    PasswordVerificationResult IPasswordHasher.VerifyHashedPassword(
        string hashedPassword, 
        string providedPassword)
    {
        return this.hasher.VerifyHashedPassword(hashedPassword, providedPassword);
    }

    string IPasswordService.HashPassword(string password)
    {
        return this.hasher.HashPassword(password);
    }

    bool IPasswordService.VerifyPassword(string hashedPassword, string testPassword)
    {
        var result = this.hasher.VerifyHashedPassword(
                         hashedPassword, 
                         testPassword);

        return result == PasswordVerificationResult.Success;
    }
}