Hate crimes are treated as separate offenses and/or add to the severity of an offense that is already criminal.
For example here is the text of the law in Illinois:
A person commits hate crime when, by reason of the actual or perceived race, color, creed, religion, ancestry, gender, sexual orientation, physical or mental disability, or national origin of another individual or group of individuals, regardless of the existence of any other motivating factor or factors, he or she commits assault, battery, aggravated assault, intimidation, stalking, cyberstalking, misdemeanor theft, criminal trespass to residence, misdemeanor criminal damage to property, criminal trespass to vehicle, criminal trespass to real property, mob action, disorderly conduct, transmission of obscene messages, harassment by telephone, or harassment through electronic communications as these crimes . . .
Damaging property is already a crime. Spray-painting anti-Semitic slurs or Nazi symbols is on a synagogue is also a hate crime in addition to be the crime of damaging property.
On the other hand, making false statements to police is a crime, but making false statements to police about a fake hate crime is not a hate crime. Should it be?