Because of a problem (created by me) in Office, I had to reinstall Office 2013.
I don’t know why, but I managed to broke the reinstall process.
The error message was: We can’t verify the signature of files required to install Office product
The logical step was: troubleshooting.
Download a new Office copy from my Office 365 account. – I got the same error.
I’ve cleaned the Temp folder, followed by a system restart. – I got the same error
I’ve used CCleaner to clear the remainings from the registry. – I got the same error
When I was about to give up and install Office 2010, I remembered that the installation is done from the Cloud.
Then I went to step 4 – clear Internet Explorer’s cookies and cache:
Internet Explorer: Tools – Internet Explorer – Delete – check Cookies and Temporary Internet Files
Now, Office was able to reinstall successfully