There are many companies that will be willing to carry out your move, us included of course.
Whoever you decide to use, the most import thing to do is to check their insurance details. Get the policy number and the name of their broker, and then look for their phone number on the net. Call them. They won't mind confirming the cover in force.
Ask removal companies, how old their vehicles are? We have seen many an overloaded tatty old Transit Luton van - sometimes road side during our many journeys up and down the Spanish and French motorways. We invest in new vehicles as should the worst happen, they are covered by the manufacturers break down assistance and warranties, allowing us to reduce any down time to a minimum. If a company can't be bothered to invest the money and commitment into their assest, ponder how they think about their customers.
Do they have an O Licence issued in the UK? (or Tareja de transportes in Spain) It means they have passed the relevant exams and have the proper back up funds in place and have to keep stringenet regular records concerning hours and saftey etc. No O licence? even if they just run a van or tow? - Walk away.
Ask how long they have been doing European removals for, what routes they use. No reputable firm will mind you asking lots of questions!
Look on internet forums to see if any one has recommended firms. Recommendations and repeat business are the key to a succesful business! Keeping advertising costs down allowing companies to keep their pricing competitive.
Cheapest is not always best. Do your checks and narrow your choice down to two or three firms. Then it really is down to gut instinct!