Is there a next step, to do? I still can’t even see the invite.

Hello Altosaxophone,

Welcome to Apple Support Communities. 

We understand that you are not able to accept an invitation to your Home. We are happy to help.

Since Apple has recently released an iOS update, please use the steps here to update your iPhone: Update your iPhone or iPad - Apple Support

"Update your iPhone or iPad wirelessly

1. Back up your device using iCloud or your computer.
2. Plug your device into power and connect to the internet with Wi-Fi.
3. Go to Settings > General, then tap Software Update.
4. If you see more than one software update option available, choose the one that you want to install.
5. Tap Install Now. If you see Download and Install instead, tap it to download the update, enter your passcode, then tap Install Now. If you don't know your passcode, learn what to do."

We recommend that both you and the Home owner, restart your iPhone using the steps here: Restart your iPhone - Apple Support

"How to restart your iPhone X, 11, 12, 13, or later

1. Press and hold either volume button and the side button until the power-off slider appears.
2. Drag the slider, then wait 30 seconds for your device to turn off. If your device is frozen or unresponsive, force restart your device.
3. To turn your device back on, press and hold the side button (on the right side of your iPhone) until you see the Apple logo."

Once both iPhone have fully restarted, have the Home owner remove the pending invitation. Once that invitation has been removed, please have them send you another invite. These steps may resolve this.

Let us know if you have further questions.

Take care.