Here is a slight refinement of Emerton's answer.

Any L-homomorphism $\phi:W_\mathbb R\rightarrow \phantom{}^LG$ defines an infinitesimal character $\lambda$ and a central character $\chi$. If $\phi(W_\mathbb R)$ is not contained in a proper Levi this defines an L-packet: the set of relative discrete series with this infinitesimal and central character (relative = discrete series modulo center). Every relative discrete series L-packet arises this way.

Now compatibility with parabolic induction determines all L-packets as follows. If $\phi(W_\mathbb R)$ is contained in a proper Levi subgroup $\phantom{}^LM$, the preceding construction defines a relative discrete series L-packet for $M$. The L-packet for $G$ is defined to be the irreducible summands of $Ind_{MN}^G(\pi_M)$ as $\pi_M$ runs over the L-packet of $M$ (for the correct choice of $N$). This induction can be done in stages: a tempered step, which is completely reducible, followed by an induction which gives unique irreducible summands. (This incorporates the "twisting" mentioned above).

As Emerton says this is canonical since we don't need L or epsilon factors to define relative discrete series L-packets. I'd like to know if anything like this, however speculative, might hold for general p-adic groups; in particular whether epsilon factors really do come to the rescue outside of GL(n) and perhaps classical groups. See Characterizing the Local Langlands Correspondence.

Here is a slight refinement of Emerton's answer.

Any L-homomorphism $\phi:W_\mathbb R\rightarrow \phantom{}^LG$ defines an infinitesimal character $\lambda$ and a central character $\chi$. If $\phi(W_\mathbb R)$ is not contained in a proper Levi this defines an L-packet: the set of relative discrete series with this infinitesimal and central character (relative = discrete series modulo center). Every relative discrete series L-packet arises this way.

Now compatibility with parabolic induction determines all L-packets as follows. If $\phi(W_\mathbb R)$ is contained in a proper Levi subgroup $\phantom{}^LM$, the preceding construction defines a relative discrete series L-packet for $M$. The L-packet for $G$ is defined to be the irreducible summands of $Ind_{MN}^G(\pi_M)$ as $\pi_M$ runs over the L-packet of $M$ (for the correct choice of $N$). This induction can be done in stages: a tempered step, which is completely reducible, followed by an induction which gives unique irreducible summands. (This incorporates the "twisting" mentioned above).

As Emerton says this is canonical since we don't need L or epsilon factors to define relative discrete series L-packets. I'd like to know if anything like this, however speculative, might hold for general p-adic groups; in particular whether epsilon factors really do come to the rescue outside of GL(n) and perhaps classical groups. See Characterizing the Local Langlands Correspondence.