How to use my non-iCloud email with iCloud calendar events

Is there a way to use my regular email address (non iCloud) to receive iCloud invites?

My contacts don't know my iCloud account, they only know my regular email address so they will send messages, calendar invites, etc to my regular email which is not my iCloud account.

Although I can accept invites to any account, there isn't the same level of integration with iOS and OS X apps that you get when the invite is sent to an iCloud account. (i.e. they don't appear in Calendar, you have to accept them from an email and it sends you off to iCloud.com) or you have to download the .ics file to add it, etc.

Is there a way to link that regular email address to my iCloud account so that invites and messages to that email are treated as though they were sent to my regular iCloud account and I can accept, reject directly from the calendar applications on either my iOS or OS X device?

Thanks!