Reciprocation is the basis of a social exchange. When someone says "hello" to you, you say "hello" in return. Someone extends his hand in greeting, you extend yours and a handshake ensues. Someone extends a dinner invitation, you later invite her for an occasion.
It's always back and forth, a two-way street, old as humankind.
All social occasions should have an exchange of some sort - which is also two-fold:
You are returning the kindness of the invitation, not necessarily dinner or the exact same type of event. Although, your exchange does include hosting your guests. Here are some ideas:
If you were invited to a celebratory event such as a wedding, birthday party, or bridal shower, there is need to reciprocate the invitation. The gift you gave is your exchange.
Nor would you need to return an invitation to an event where you paid something. For instance, a charity event where you gave a donation, or a music festival for which you paid for your own ticket.
A return invitation is also not necessary when you are invited to sit at the head table at an event - e.g. the president of your university, a government official, etc.
One sticky situation that sometimes occurs is that after having dinner with someone (or a couple), you decide you really don't care to spend more time getting to know them or engaging in more conversation. In this case, it is not required that you return that dinner invitation - although it would be a lovely gesture if you can stomach it.
However, in each of the above examples, a thank-you is absolutely appropriate. And when that thank-you is in the form of a handwritten note, it is ideal and most appreciated. Send these as soon as possible!
When someone invites you to her home for dinner, she has gone to the trouble of setting up an occasion to enjoy your company. Your time together feels special and creates connections.
By reciprocating, you are extending yourself to continue the relationship. It is also fun to see another person enjoy an occasion you created. This is one of the pleasures of being a host.
What if someone doesn't reciprocate your dinner invitation? Perhaps he doesn't know that he should. A little forgiveness and understanding go a long way. And it is perfectly okay to invite someone to your home again and again if you want, especially if everyone enjoys themselves.
Just like all etiquette guidelines, reciprocity is situational and contextual. You might not have the means to exchange in kind, and this is okay. Simple is better than nothing. Be practical and authentic in the invitations you extend. They'll love you for who you are!