Satanic and ridiculous this riddle may seem,
Unify the beginnings to find out its theme.
Repress a disaster; don't look everywhere,
Right words exhibit a trait that they share.
Only these dodgy tertiary ones count,
Unveil the relevant parts you've now found.
Now a chihuahua will show you the way,
Dogs are deceptive, but this one's OK.
Eliminate the arbitrary and useless kind,
Demonic, but elegant, the solution you'll find.
As some of the answers below have already correctly pointed out, only some of the words in the riddle are actually important. The theme tells you something about the trait they share...
Only one of the answers below has the correct "tertiary" words. Look for the trait there.