In practice, Alexa and Cortana will first have to be enabled as a skill on each competing platform. For example, the Cortana skill will have to be installed on an Amazon Echo, while the Alexa skill will have to be installed on a Windows 10 PC. From there, you have to do a small amount of verbal gymnastics to access the respective AI assistant platforms.
On a Windows 10 PC or Harman Kardon Invoke speaker, you can say "Hey Cortana, open Alexa." After issuing that command, you can access the entire suite of Alexa smart home capabilities or any number of Alexa skills (like calling an Uber). From the Amazon hardware side, you can use your Amazon Echo Show, for example, to say "Alexa, open Cortana". At that time, you'll then be able to access calendar event and other productivity-related tasks within the Windows and Office ecosystems.
It's not exactly a pairing of equals given Amazon's domination of the AI speaker market and its wealth of Alexa skills. However, Microsoft does bring a lot to the table, giving Amazon Echo users the ability to access their email and calendar details with voice commands.
As we previously stated, this AI crossover is currently only available in the United States and there was no timeline given for when it will be available in other world markets.