Plot spectrum of $X(-k)$
FFT (DFT) is defined as


  $$\mathsf{DFT}\{x[n]\}=X[k]=\sum_{n=0}^{N-1}x[n]e^{-jk2\pi n/N}$$


Hence, $X[-k]$ is 
$$X[-k]=\sum_{n=0}^{N-1}x[n]e^{jk2\pi n/N}$$

By investigating the formula of inverse DFT (and a quick change of variable $n\leftrightarrow k$ for more clarity),


  $$\mathsf{IDFT}\{X[k]\}=x[n]=\frac1N\sum_{k=0}^{N-1}X[k]e^{jk2\pi n/N}\Rightarrow x[k]=\frac1N\sum_{n=0}^{N-1}X[n]e^{jk2\pi n/N}$$


We can see that
$$X[-k]=N\cdot\mathsf{IDFT}\{x[n]\}$$
So instead of calculating the FFT and then converting the result, you can directly calculate $X[-k]$ by taking the IFFT.