It's hard to tell where your mail is failing, what services/etc are you using? Any error messages?
A few things to consider:
1) Make sure you have an MX record pointing to a valid A record (IE MX record:
mail.yourdomain.com and A record:
mail.yourdomain.com points to 188.8.131.52
2) Check your mail configurations: Most SMTP servers are not configured as open servers, as a spammer could use your mail server to send their mails. Look at 'mynetworks' or allowed hosts and make sure whoever is sending mail to your server is in this list.
3) Check the logs: Turn on verbose logging for your MTA. With postfix, edit /etc/postfix/master.cf and append " -v" at the end of smtpd (smtpd -v) and by default is logged to /var/log/messages
As for reverse DNS, just make sure that the hostname of your mail server is a FQDM (Fully Qualified Domain Name) and has the appropriate A records assigned to it. For ex. if your server's public address is at 184.108.40.206, and the hostname of your machine is mail.mydomain.com, then your A record should be mail.mydomain.com pointing to 220.127.116.11. in /etc/hosts you should have an entry with the same name (IE 127.0.0.1 mail.mydomain.com localhost) or (18.104.22.168 mail.mydomain.com)