EU international rules are quite complicated when it is about a social security matter.

Under Community rules, workers moving within the European Union must be subject to a single social security legislation
The social security scheme applicable to those who for reasons of work move from one Member State to another is, generally speaking, that established by the legislation of the Member State of new employment.
Indeed, if you want to work temporarily abroad, it would be easier to remain “as if” you was still depending of YOUR national security scheme.
The Council Regulation (EEC) N° 1408/71 of 14 June 1971 on the application of social security schemes to employed persons, to self-employed persons and to members of their families moving within the European Union grants the option to maintain the social security scheme of the Member State in which the undertaking normally operates (the “posting State”) whenever the worker concerned is sent by that undertaking to another Member State (the “State of Employment”) for a period of time which from the outset is limited (generally a maximum of 12 months) and under specific conditions.
If a worker goes abroad, in one or more countries, and that he wants to maintain his social security scheme, he should apply for Form E101. This document is issued to persons who are employed or self-employed, working on a temporary basis in another EU state for a maximum of 12 months.
Nevertheless, it will be possible to renew this demand with form E-102. This document is issued for an extension of up to a further 12 months sometimes up to 5 years, but this is a maximum.
BE CAREFUL: this does NOT mean that you will be allowed to use your European Health Insurance Card (blue card) as if you were abroad for a few weeks; you will have to go to Spanish doctors if you decide to live in Tenerife.
For tax purposes, you should consider in which country you will your tax bill; the member States of the European Union signed Double Taxation Avoidance Treaties.
Generally, you will have to pay taxes in the country where you are working (so, abroad); this will be the case if you work at least 183 days in this country. Eventually, within big companies, you will benefit from a “salary split”.
Stéphane MERCIER
Tax expert and economist; international consultant.
Administrator of EXPERTOS DE TENERIFE S.L.