If Wordpress is passing the user to Centrify to login (SAML), when the user comes back to Wordpress the .ASPXAUTH token is already set as a cookie in the browser. You cant access it in code, but if you make browser (CORS) calls to Centrify API's, the browser will automatically pass the cookie to Centrify with out you needing to manage it or pass it manually. These calls would need to be made in the browser (javascript), not from the server (php).
You will need to go into your Centrify admin portal, navigate to settings > Authentication > Security Settings > Specify trusted DNS domains for API calls and add the domain of the site that will be calling the Centrify API's. This is to trust the domain for CORS.
An easy test is to add a small amount of code to the browser code that does a simple post to https://tenant.my.centrify.com/security/whoami. No JSON needs to be passed to this call. This will simply respond telling you if we see the user as authenticated.
Please let me know if you have any other questions and do not hesitate to reach out to devsupport@centrify.com.
Thank you,
Nick Gamb
Sr. Manager, Developer Relations and Product Management
Centrify Corporation