Using a table of Carmichael numbers up to $10^{16}$, there are $34971$ pairs $(c-2,c)$ where $c$ is a Carmichael number and $c-2$ is prime but only $204$ pairs $(c,c+2)$ with $c+2$ prime.
Is there some theoretically reason for this striking asymmetry ($99.42\%$ vs. $0.58\%$)?
The MSE question Why are Carmichael Numbers less common with an arithmetic progression seems somehow related, but I cannot see a direct connection.
Looking at the Carmichael numbers up to $10^8$, the list is dominated by numbers that are $1 \bmod 3$. Out of $ 255 $ numbers, $ 243 $ are $1\bmod 3$ - over $ 95\% $, and the trend is towards a higher proportion.
This is fairly natural, since for a given Carmichael $c$, we need $c{-}1$ to be divisible by each of its prime factor totients, and among all these multiplying components the likelihood that $3$ is included as one of the multipliers is large, since $3$ is not often a factor of $c$ itself.
An aside: Carmichael numbers divisible by $3$ are uncommon but at least one of them - $656601$ - gives us a prime at both $c{-}2$ and $c{+}2$. Triplets? Better: $c{-}4$ is also a prime, so quadruplets!
