How can I see the specific value of the sql_mode?

It's only blank for you because you have not set the sql_mode. If you set it, then that query will show you the details:

mysql> SELECT @@sql_mode;
+------------+
| @@sql_mode |
+------------+
|            |
+------------+
1 row in set (0.00 sec)

mysql> set sql_mode=ORACLE;
Query OK, 0 rows affected (0.00 sec)

mysql> SELECT @@sql_mode;
+----------------------------------------------------------------------------------------------------------------------+
| @@sql_mode                                                                                                           |
+----------------------------------------------------------------------------------------------------------------------+
| PIPES_AS_CONCAT,ANSI_QUOTES,IGNORE_SPACE,ORACLE,NO_KEY_OPTIONS,NO_TABLE_OPTIONS,NO_FIELD_OPTIONS,NO_AUTO_CREATE_USER |
+----------------------------------------------------------------------------------------------------------------------+
1 row in set (0.00 sec)

You can also try this to determine the current global sql_mode value:

SELECT @@GLOBAL.sql_mode;

or session sql_mode value:

SELECT @@SESSION.sql_mode;

I also had the feeling that the SQL mode was indeed empty.

You need to login to your mysql terminal first using mysql -u username -p password

Then use this:

SELECT @@sql_mode; or SELECT @@GLOBAL.sql_mode;

output will be like this:

STRICT_TRANS_TABLES,STRICT_ALL_TABLES,NO_ZERO_IN_DATE,NO_ZERO_DATE,ERROR_FOR_DIVISION_BY_ZERO,TRADITIONAL,NO_AUTO_CREATE_USER,NO_ENGINE_SUB

You can also set sql mode by this:

SET GLOBAL sql_mode=TRADITIONAL;