Since I believe HC created a new wallet for me when I registered and since they provided me with my secret keywords, I assume they can spend my balance. Maybe I am mistaken?
From the HC wallet page:
"Ownership and control over the private key is the root of user control over all funds associated with the corresponding Bitcoin Cash address. The private key is used to create signatures that are required to spend bitcoins by proving ownership of funds used in a transaction. The private key must remain secret at all times, because revealing it to third parties is equivalent to giving them control over the bitcoins secured by that key. The private key must also be backed up and protected from accidental loss, because if it’s lost it cannot be recovered and the funds secured by it are forever lost, too."