Can you combine types of cat litter?

At present we have a feline houseguest. He is used to using the clumping sawdust-type litter. The supply recently ran out before his person brought more, so we resorted to using some of the Jonny Cat kept in the garage for occasional oil spills.

The Jonny Cat seems to be less odorous, and it also does not get tracked all over the house like the sawdust, which clings to the cat’s fur; but it doesn’t have the helpful clumping feature. The cat’s owner does not like him to use the clay litter because she is worried that the dust will be bad for him; he has several health conditions. I want to respect her preferences, but the cat is staying in our house and we are living with the smell and the maintenance without enjoying much of a relationship with the cat. It would be nice to keep this arrangement as low-impact as possible.

Would it be a bad thing to combine the two types of litter to get the benefits of both? Has any of you cat people tried this? Is there a down side?