Can I replace two Siemens QP type, handle tied, single pole 15 amp breakers with a tandem QT type in the same configuration?
NO. Your current configuration puts each hot on a different leg, resulting in 240 volts (120+120). The tandem breaker you’re proposing puts both hots on the same 120 volt leg. This will result in zero volts across the hots. If this were a multi-wire branch circuit, you’d end up with double current on the neutral. 

What you could use is a compatible quad breaker (4 breakers in the space for two) with appropriate handle ties to produce two 240 volt circuits or one 240 and two 120s.