The Cross isn't an ornament, mere symbol. It's the mystery of God's love, that He died for our sins.
But dogs, which have no ability to sin nor moral conscience, do not have an ability to reject Jesus.
Sin is the despairing refusal to find your deepest identity in your relationship and service to God.
I will but confess the sins of my green cloak to my grey friar's frock, and all shall be well again.