When you get employed at the University of Southern Denmark while living in Sweden, there are some practical issues you need to settle before starting your employment, e.g. social security, taxation, etc.
To receive your salary you need to apply for a tax card and a personal tax number using this online form and send it to SKAT. You also have to open a Danish bank account (NEM-konto). If you want to apply for the special tax scheme for researchers you need to fill out a form in cooperation with your department. You can find the form here.
If you apply for the special tax scheme for researchers you can send both of the forms directly to the salary office. Read more about the special tax scheme for researchers.
When you live in Sweden and work in Denmark you are covered by social security in Denmark. Please use this online form to apply for a special Health Card. The special health card is your proof of the right to health care in Denmark.
If you also work in Sweden for another employer you will still be covered by Danish social security as long as your work in Sweden does not exceed 25 % of your total working hours. Read more about work in more than one country.
If you want to be covered by unemployment insurance you have to sign up in an “A-kasse” in Denmark, but you will receive unemployment benefits from Sweden if you lose your job. Read more about the rules here.
At oresunddirektbusiness.dk you can read about the relevant rules and find a checklist on what to remember in connection with your employment.
At norden.org you can find supplementary information.