Consider the $ [[4,2,2]] $ stabilizer code with stabilizer generators $ XXXX,ZZZZ $. Logical $ X $ on the first qubit is $$ XIXI $$ Logical $ Z $ on the first qubit $$ ZZII $$ Logical $ X $ on the second qubit is $$ XXII $$ Logical $ Z $ on the second qubit is $$ ZIZI $$ It is also the case that $ SWAP_{2,3} $ implements logical SWAP (this isn't technically transversal, but it is a code automorphism).
For any single qubit Clifford gate $ C $ the physical gate $ C^{\otimes 4} $ preserves the code space, implementing some logical operation. For example $ H^{\otimes 4} $ and $ P^{\otimes 4} $ both implement logical gates. The specific gates are described on page 3 of https://arxiv.org/abs/1610.03507 just before Figure 3. See also page 165.
Are there any other transversal gates besides these? Really my question is is there a good reference listing (or claiming to list) all transversal gates of the $ [[4,2,2]] $ code?