Twilio fax - How to manage status responses in Wordpress in PHP?

I want to send and manage faxes using PHP only (no JavaScript or cron, if possible) through Twilio from my linux+Apache+Wordpress site. (Development is on a Wordpress site through my localhost using ngrok.)

I have been able to set up a Wordpress page (template) to send faxes using Twilio's fax API. After the fax is sent, though, Twilio is replying with JSON in a POST response (I think), and expecting my site to respond in some way. I have a callback URL in place, but this seems to be an asynchronous call, which I don't know how to handle though Wordpress. (I want to stay within Wordpress site if possible for security and convenience.)

I have no experience with managing this type of communication between servers; I have been reading the Twilio docs, but I think I am making a fundamental mistake somewhere...I get the gist of what needs to be done, but not how it works.

How does managing Twilio's asynchronous calls work using PHP within Wordpress?

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