How does this quantity transform under time reversal and inversion operation respectively?

Suppose $$\psi_{nk}=e^{ikr}u_{nk}$$ is the Bloch function of a periodic Hamiltonian $$H(r)=H(r+R)$$, where $$H(r)\psi_{nk}=\varepsilon_{nk}\psi_{nk}$$ and $$H(k)=e^{-ikr}H(r)e^{ikr}$$.

What would the following quantity change under the time-reversal and inversion operation respectively and why?

$$\vec{L}(k)=i\int \mathrm{d}r \nabla_k u_{nk}^*\times\left[H(k)-\varepsilon_{nk}\right]\nabla_k u_{nk}$$

I have no clue how to tackle this, I don't know if I am going to change $$u_{nk}$$ or to change $$H(k)$$ and $$\varepsilon_{nk}$$ or change both, and I am not sure how to change them. Please give a detailed answer if you know how to do this.