Employment Ethics and Legal Implications
I believe that employment is a commitment between the employer and employee, which should adhere to ethical guidelines. Unfortunately, this practice has diminished, leading to regulation through various labor laws and governmental interventions.
From your description, it appears your employer has not been "ethical," as they did not issue you a letter of appointment. Similarly, you did not provide notice before resigning from the company. Both actions seem far from "legal," and if brought before a legal forum, could result in significant penalties, leaving a "permanent scar" on both parties.
It seems best to resolve this amicably through mutual discussions and a settlement process. I hope your ex-employer will be reasonable enough to at least pay you the "due salary" for the period you worked. A letter of experience from such a company may not be worthwhile, and avoiding bitterness might be beneficial.
This situation should serve as an "eye-opener" for future employments.