How to tell if breaker box can handle another tandem breaker?
Your box predates the UL requirements for a feature known as Circuit Total Limitation (CTL).  This feature restricts where double-stuff/tandem breakers can be installed in the panel to prevent it from exceeding its designed circuit capacity, which was restricted to a maximum of 42 circuits by the NEC (i.e. nobody could make a panelboard for more than 42 circuits).  However, the 2008 NEC lifted the 42 circuit restriction on panelboards.

Practically speaking, I would go ahead and install the new double-stuff/tandem breaker provided that there is sufficient remaining wiring space in the panelboard gutters to accommodate the extra wire.  You'll need a QO2020 (not a QOT2020, which won't fit) breaker for this.