Raising the index of the stress energy tensor

Question

By considering the variation of a lagrangian which has no explicit space time dependence under an arbitrary spacetime translation, $a^{\mu}$ I've seen that we can express the variation in the lagrangian as $$\delta \mathcal{L}=a^{\nu}\partial_{\mu}\mathcal{L}$$$$=a^{\mu}\partial_{\mu}(\delta^{\nu}_{\mu}\mathcal{L})$$$$=a^{\mu}\partial_{\mu}J^{\nu}_{\mu}$$. Using Noether's theorem we can therefore find the four conserved Noether currents in the form of the stress energy tensor: $$T^{\nu}_{\mu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi)}\partial_{\mu}\phi-J^{\nu}_{\mu}$$ I'm comfortable up to this point, however I don't understand: (a) why we can raise the index to give: $$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\nu}\phi)}\partial^{\mu}\phi-g^{\mu\nu}\mathcal{L}$$ and (b) this tensor isn't obviously symmetric to me.