Then I found TestDisk and ran it from an Ubuntu Live USB.
It gave me this:
When listing files using the P command all files seemed to be right there (apart from the Linux Swap of course). But I can not figure out how to correctly specify what partition shall be Primary, Logical etc. However I seemed to remember that the NTFS partitions all were Primary once upon a time and that the two Linux partitions were two logical partitions in an extended partition. However it seems to be hard to specify that configuration. At least I am unable to do that. Using left/right arrows do not allow me to specify Extended partition. Someone that know how to correctly set the partition table?TestDisk 7.0, Data Recovery Utility, April 2015
Christophe GRENIER <email@example.com>
Disk /dev/sda - 120 GB / 111 GiB - CHS 14593 255 63
Partition Start End Size in sectors
>D HPFS - NTFS 0 32 33 63 221 30 1024000
D HPFS - NTFS 63 221 31 7549 59 62 120252416
D HPFS - NTFS 7549 59 63 7648 236 12 1601536
D Linux 7649 46 14 13555 108 57 94883840
D Linux Swap 13555 108 58 14593 66 1 16672768
Structure: Ok. Use Up/Down Arrow keys to select partition.
Use Left/Right Arrow keys to CHANGE partition characteristics:
*=Primary bootable P=Primary L=Logical E=Extended D=Deleted
Keys A: add partition, L: load backup, T: change type, P: list files,
Enter: to continue
NTFS, blocksize=4096, 524 MB / 500 MiB
BTW. Before doing any changes I plan to mirror the drive so that I can go back later on..