Remote Connectivity Analyzer is a free connectivity test platform for the cloud-based service. It tests the availability of the required Office 365 SSO service endpoint for expected behavior by acting on those services from the Internet.
Is simply the best way to check issues with Office365, and to be honest any Exchange platform.
It has many features as show in the below screenshots
To verify that inbound mail, Exchange ActiveSync, Autodiscover, Outlook Anywhere (RPC/HTTP), and other connections are properly configured, you should consider using the Exchange Remote Connectivity Analyzer. The Remote Connectivity Analyzer is a Web-based tool that is designed to help you troubleshoot connectivity issues by testing your Exchange Online and on-premises configuration. You may want to become familiar at this tool in advance of your deployment to understand its capabilities and usage. For more information about the tool, see this TechNet article.
Additionally, if you already have ActiveSync, Outlook Anywhere, and Autodiscover enabled in your on-premises environment we highly recommend you run through the tests provided with the Remote Connectivity Analyzer to ensure your on-premises environment is functioning appropriately.